💡Alright, so I've got this trigonometry problem here, and I need to figure out in which quadrant the angle α lies based on the given conditions. The problem says:"If angle α satisfies the conditions sin 2α < 0 and cosα - sinα < 0, then α is in the ()"And the options are:A: first quadrantB: second quadrantC: third quadrantD: fourth quadrantHmm, okay. Let me break this down step by step.First, I know that the sine and cosine functions have specific signs in each quadrant. Maybe I can use that to figure out where α is. Let me recall:- In the first quadrant (0 to 90 degrees), both sine and cosine are positive.- In the second quadrant (90 to 180 degrees), sine is positive and cosine is negative.- In the third quadrant (180 to 270 degrees), both sine and cosine are negative.- In the fourth quadrant (270 to 360 degrees), sine is negative and cosine is positive.Okay, so that's a good start. Now, the first condition is sin 2α < 0. Let me think about what that tells me.I remember that sin(2α) is just another way of writing the sine of double the angle α. So, if sin(2α) is negative, that means that 2α must be in a quadrant where sine is negative. From what I remember, sine is negative in the third and fourth quadrants. So, 2α must be in either the third or fourth quadrant.But wait, 2α is double the angle α. So, if α is in a certain quadrant, 2α could be in a different quadrant. Let me think about this.Let's consider the possible quadrants for α:1. If α is in the first quadrant (0 to 90 degrees), then 2α would be between 0 and 180 degrees. So, 2α would be in the first or second quadrant. But in both these quadrants, sine is positive. So, sin(2α) would be positive, which contradicts the condition sin(2α) < 0. Therefore, α cannot be in the first quadrant.2. If α is in the second quadrant (90 to 180 degrees), then 2α would be between 180 and 360 degrees. So, 2α would be in the third or fourth quadrant. In both these quadrants, sine is negative. So, sin(2α) would be negative, which satisfies the first condition. So, α could be in the second quadrant.3. If α is in the third quadrant (180 to 270 degrees), then 2α would be between 360 and 540 degrees. But since angles are periodic every 360 degrees, 360 to 540 is the same as 0 to 180 degrees. So, 2α would be in the first or second quadrant again, where sine is positive. So, sin(2α) would be positive, which doesn't satisfy the condition. Therefore, α cannot be in the third quadrant.4. If α is in the fourth quadrant (270 to 360 degrees), then 2α would be between 540 and 720 degrees. Again, subtracting 360, that's 180 to 360 degrees. So, 2α would be in the third or fourth quadrant, where sine is negative. So, sin(2α) would be negative, satisfying the first condition. Therefore, α could also be in the fourth quadrant.So, from the first condition alone, α could be in either the second or fourth quadrant.Now, let's look at the second condition: cosα - sinα < 0. This simplifies to cosα < sinα. So, in which quadrants is cosine less than sine?Let me think about the unit circle again.In the first quadrant, both sine and cosine are positive, but cosine is greater than sine for angles between 0 and 45 degrees, and sine is greater than cosine for angles between 45 and 90 degrees. So, depending on the angle, cosα could be less than sinα.But wait, we already determined that α can't be in the first quadrant because of the first condition. So, maybe I don't need to consider that.In the second quadrant, sine is positive and cosine is negative. So, cosα is negative, and sinα is positive. So, cosα < sinα because a negative number is always less than a positive number. So, in the second quadrant, cosα - sinα would be negative, satisfying the second condition.In the fourth quadrant, sine is negative and cosine is positive. So, cosα is positive, and sinα is negative. So, cosα - sinα would be positive minus negative, which is positive plus positive, so it's positive. Therefore, cosα - sinα would be greater than 0, which doesn't satisfy the second condition.Wait, that seems contradictory. Let me double-check.If α is in the fourth quadrant, then cosα is positive and sinα is negative. So, cosα - sinα would be positive minus negative, which is positive plus positive, so it's positive. Therefore, cosα - sinα > 0, which does not satisfy the condition cosα - sinα < 0.Therefore, α cannot be in the fourth quadrant.So, putting it all together, from the first condition, α could be in the second or fourth quadrant. From the second condition, α cannot be in the fourth quadrant. Therefore, α must be in the second quadrant.Wait, but let me think again. Is there a possibility that in the second quadrant, cosα - sinα is negative?Yes, because in the second quadrant, cosα is negative and sinα is positive, so cosα - sinα is negative minus positive, which is negative. So, that's correct.But hold on, in the second quadrant, is cosα always less than sinα? Or is it that cosα is negative and sinα is positive, so cosα is definitely less than sinα?Yes, because cosα is negative and sinα is positive, so cosα - sinα is negative. So, that's why the second condition is satisfied.But wait, in the second quadrant, is there a specific range where cosα - sinα is negative?Actually, in the second quadrant, for all angles, cosα is negative and sinα is positive, so cosα - sinα is always negative. Therefore, the second condition is always satisfied in the second quadrant.But in the fourth quadrant, as I thought earlier, cosα is positive and sinα is negative, so cosα - sinα is positive minus negative, which is positive. So, it doesn't satisfy the second condition.Therefore, combining both conditions, α must be in the second quadrant.Wait, but let me think again about the first condition. If α is in the second quadrant, then 2α would be in the third or fourth quadrant.But if α is in the second quadrant, say α is 120 degrees, then 2α is 240 degrees, which is in the third quadrant, where sine is negative. So, sin(2α) is negative, which satisfies the first condition.If α is 150 degrees, then 2α is 300 degrees, which is in the fourth quadrant, where sine is negative. So, sin(2α) is negative, which also satisfies the first condition.So, in both cases, whether α is in the second quadrant, 2α is in the third or fourth quadrant, where sine is negative.Therefore, both conditions are satisfied when α is in the second quadrant.But wait, what if α is in the fourth quadrant? Let's take α as 300 degrees. Then, 2α is 600 degrees, which is equivalent to 240 degrees (600 - 360 = 240), which is in the third quadrant, where sine is negative. So, sin(2α) is negative, satisfying the first condition.But then, cosα - sinα: cos(300) is 0.5, sin(300) is -√3/2. So, 0.5 - (-√3/2) = 0.5 + √3/2, which is positive. So, cosα - sinα > 0, which doesn't satisfy the second condition.Therefore, even though 2α is in the third quadrant (satisfying sin(2α) < 0), the second condition is not satisfied because cosα - sinα is positive.Therefore, α cannot be in the fourth quadrant.So, putting it all together, the only quadrant where both conditions are satisfied is the second quadrant.Wait, but let me think about another angle in the second quadrant. Let's take α = 135 degrees. Then, 2α = 270 degrees. Sin(270) is -1, which is negative, so sin(2α) < 0 is satisfied.Now, cos(135) is -√2/2, sin(135) is √2/2. So, cosα - sinα = (-√2/2) - (√2/2) = -√2, which is less than 0. So, the second condition is satisfied.Another example: α = 150 degrees. 2α = 300 degrees. Sin(300) is -√3/2, which is negative.Cos(150) is -√3/2, sin(150) is 1/2. So, cosα - sinα = (-√3/2) - (1/2) = (-√3 - 1)/2, which is negative.So, both conditions are satisfied.But wait, what if α is 90 degrees? Then, 2α is 180 degrees. Sin(180) is 0, which is not less than 0. So, α = 90 degrees is not acceptable.Similarly, α = 180 degrees: 2α = 360 degrees. Sin(360) is 0, which is not less than 0.So, α must be strictly between 90 and 180 degrees, not including the endpoints.Therefore, the angle α is in the second quadrant.Wait, but let me think about another angle in the second quadrant, say α = 100 degrees. Then, 2α = 200 degrees. Sin(200) is negative because 200 degrees is in the third quadrant. So, sin(2α) < 0 is satisfied.Cos(100) is negative, sin(100) is positive. So, cosα - sinα is negative - positive, which is negative. So, the second condition is satisfied.Another example: α = 170 degrees. 2α = 340 degrees. Sin(340) is negative because 340 degrees is in the fourth quadrant. So, sin(2α) < 0 is satisfied.Cos(170) is negative, sin(170) is positive. So, cosα - sinα is negative - positive, which is negative. So, the second condition is satisfied.Therefore, in all these cases, both conditions are satisfied when α is in the second quadrant.But just to be thorough, let me check an angle in the third quadrant. Let's say α = 210 degrees. Then, 2α = 420 degrees, which is equivalent to 60 degrees (420 - 360 = 60). Sin(60) is positive, so sin(2α) > 0, which doesn't satisfy the first condition. So, α cannot be in the third quadrant.Similarly, in the first quadrant, α = 60 degrees. 2α = 120 degrees. Sin(120) is positive, which doesn't satisfy the first condition.In the fourth quadrant, as I checked earlier, even though sin(2α) < 0, cosα - sinα > 0, so it doesn't satisfy the second condition.Therefore, the only quadrant where both conditions are satisfied is the second quadrant.Wait, but let me think about the angle α = 225 degrees. That's in the third quadrant. 2α = 450 degrees, which is equivalent to 90 degrees. Sin(90) is 1, which is positive, so sin(2α) > 0, which doesn't satisfy the first condition.Similarly, α = 300 degrees, which is in the fourth quadrant. 2α = 600 degrees, which is equivalent to 240 degrees. Sin(240) is negative, so sin(2α) < 0 is satisfied. But cos(300) - sin(300) = 0.5 - (-√3/2) = 0.5 + √3/2 > 0, so the second condition is not satisfied.Therefore, the only possible quadrant is the second quadrant.Wait, but let me think about the angle α = 135 degrees again. 2α = 270 degrees. Sin(270) is -1, which is negative. Cos(135) is -√2/2, sin(135) is √2/2. So, cosα - sinα = -√2/2 - √2/2 = -√2 < 0. So, both conditions are satisfied.Another example: α = 120 degrees. 2α = 240 degrees. Sin(240) is -√3/2 < 0. Cos(120) is -0.5, sin(120) is √3/2. So, cosα - sinα = -0.5 - √3/2 < 0.So, yes, both conditions are satisfied in the second quadrant.Therefore, the answer should be the second quadrant, which is option B.

💡Alright, so I have this problem here where I need to show that for a set ( S ) of ( n ) points in the plane, with no three points being collinear, and for every point ( P ) in ( S ), there's a real number ( r ) such that at least ( k ) points are at distance ( r ) from ( P ). I need to prove that ( k < frac{1}{2} + sqrt{2n} ).Hmm, okay. Let me try to unpack this. First, the set ( S ) has ( n ) points, and no three are collinear, which probably helps avoid some degenerate cases. Then, for each point ( P ), there's some circle centered at ( P ) that contains at least ( k ) other points on its circumference. So, each point has at least ( k ) co-radial points.I think this is related to something called the "Erdős distance problem," but I'm not entirely sure. Maybe it's about counting distances or something similar. Let me think. If each point has at least ( k ) points at the same distance, that might mean there are a lot of repeated distances, which could be connected to some combinatorial geometry principles.Wait, the problem mentions triplets ( (X, Y, Z) ) such that ( XY = YZ ). That seems important. So, if I fix two points ( X ) and ( Z ), the point ( Y ) that makes ( XY = YZ ) lies on the perpendicular bisector of ( XZ ). Since no three points are collinear, there can be at most two such points ( Y ) for each pair ( XZ ). So, the number of such triplets ( Q ) is bounded by ( 2 cdot binom{n}{2} ), which is ( 2n(n - 1) ).On the other hand, for each point ( Y ), there are at least ( k ) points at some distance ( r ) from ( Y ). So, choosing two points ( X ) and ( Z ) from these ( k ) points, we get ( binom{k}{2} ) pairs. Since there are ( n ) points ( Y ), the total number of triplets ( Q ) is at least ( n cdot frac{k(k - 1)}{2} ).Putting these together, we have:[ n cdot frac{k(k - 1)}{2} leq 2n(n - 1) ]Simplifying this inequality:[ frac{k(k - 1)}{2} leq 2(n - 1) ][ k(k - 1) leq 4(n - 1) ]Hmm, this looks like a quadratic in ( k ). Let me write it as:[ k^2 - k - 4n + 4 leq 0 ]To solve this quadratic inequality, I can use the quadratic formula. The roots of the equation ( k^2 - k - 4n + 4 = 0 ) are:[ k = frac{1 pm sqrt{1 + 16n - 16}}{2} ][ k = frac{1 pm sqrt{16n - 15}}{2} ]Since ( k ) is a natural number, we take the positive root:[ k leq frac{1 + sqrt{16n - 15}}{2} ]But this doesn't seem to match the desired inequality ( k < frac{1}{2} + sqrt{2n} ). Maybe I made a mistake in my calculations. Let me check.Wait, when I simplified ( k(k - 1) leq 4(n - 1) ), I should have considered completing the square or another method. Let me try completing the square:[ k^2 - k leq 4n - 4 ][ k^2 - k + frac{1}{4} leq 4n - 4 + frac{1}{4} ][ left(k - frac{1}{2}right)^2 leq 4n - frac{15}{4} ]Taking square roots:[ k - frac{1}{2} leq sqrt{4n - frac{15}{4}} ][ k leq frac{1}{2} + sqrt{4n - frac{15}{4}} ]Hmm, this is closer but still not the same as the desired inequality. Maybe I need to approximate or find a bound for the square root term.Notice that ( sqrt{4n - frac{15}{4}} ) is less than ( sqrt{4n} ), which is ( 2sqrt{n} ). But the desired bound is ( sqrt{2n} ), which is smaller. So perhaps my approach isn't tight enough.Alternatively, maybe I should consider that ( sqrt{4n - frac{15}{4}} ) is approximately ( 2sqrt{n} - frac{15}{8sqrt{n}} ), but that might complicate things.Wait, perhaps I should go back to the inequality before completing the square:[ k^2 - k leq 4n - 4 ]Let me rearrange it:[ k^2 - k + frac{1}{4} leq 4n - 4 + frac{1}{4} ][ left(k - frac{1}{2}right)^2 leq 4n - frac{15}{4} ]So,[ left(k - frac{1}{2}right)^2 < 2n ]Because ( 4n - frac{15}{4} ) is less than ( 2n ) for ( n geq 2 ). Wait, is that true?Let me check:[ 4n - frac{15}{4} < 2n ][ 2n < frac{15}{4} ][ n < frac{15}{8} ]But ( n ) is at least 3 since we have points and no three are collinear. So, this approach might not hold. Maybe I need a different way to bound.Alternatively, perhaps I should use the inequality ( k^2 - k leq 4n - 4 ) and note that ( k^2 < 4n ), so ( k < 2sqrt{n} ). But the desired bound is tighter: ( sqrt{2n} ).Wait, maybe I can use the fact that ( k^2 - k leq 4n - 4 ) implies ( k^2 < 4n ), so ( k < 2sqrt{n} ). But the problem wants ( k < frac{1}{2} + sqrt{2n} ). Since ( sqrt{2n} ) is less than ( 2sqrt{n} ), my current bound isn't sufficient.Perhaps I need to refine the initial counting. Let me think again about the number of triplets ( Q ).Each point ( Y ) has at least ( k ) points at distance ( r_Y ) from it. So, for each ( Y ), there are ( binom{k}{2} ) pairs ( (X, Z) ) such that ( XY = YZ = r_Y ). Therefore, the total number of such triplets is ( n cdot binom{k}{2} ).But each such triplet is counted multiple times. Specifically, for each pair ( (X, Z) ), there can be at most two points ( Y ) such that ( XY = YZ ). So, the total number of triplets ( Q ) is at most ( 2 cdot binom{n}{2} ).Thus,[ n cdot frac{k(k - 1)}{2} leq 2 cdot frac{n(n - 1)}{2} ][ frac{k(k - 1)}{2} leq n - 1 ][ k(k - 1) leq 2(n - 1) ]Ah, I see! Earlier, I had a factor of 4, but it should be 2 because ( 2 cdot binom{n}{2} = 2 cdot frac{n(n - 1)}{2} = n(n - 1) ). Wait, no, actually:Wait, ( Q leq 2 cdot binom{n}{2} ) because for each pair ( (X, Z) ), there are at most two ( Y )'s. So,[ Q leq 2 cdot frac{n(n - 1)}{2} = n(n - 1) ]But earlier, I had:[ Q geq n cdot frac{k(k - 1)}{2} ]So,[ n cdot frac{k(k - 1)}{2} leq n(n - 1) ][ frac{k(k - 1)}{2} leq n - 1 ][ k(k - 1) leq 2(n - 1) ]This is better. Now, let's solve this quadratic inequality:[ k^2 - k - 2n + 2 leq 0 ]Using the quadratic formula:[ k = frac{1 pm sqrt{1 + 8n - 8}}{2} ][ k = frac{1 pm sqrt{8n - 7}}{2} ]Taking the positive root:[ k leq frac{1 + sqrt{8n - 7}}{2} ]Hmm, still not exactly ( frac{1}{2} + sqrt{2n} ), but closer. Let me see:[ frac{1 + sqrt{8n - 7}}{2} = frac{1}{2} + frac{sqrt{8n - 7}}{2} ]Now, ( sqrt{8n - 7} ) is approximately ( sqrt{8n} = 2sqrt{2n} ), but that's not helpful. Wait, actually,[ sqrt{8n - 7} = sqrt{4 cdot 2n - 7} = 2sqrt{2n - frac{7}{4}} ]But that's not helpful either. Maybe I can bound ( sqrt{8n - 7} ) from above.Note that ( 8n - 7 < 8n ), so ( sqrt{8n - 7} < sqrt{8n} = 2sqrt{2n} ). Therefore,[ k < frac{1}{2} + frac{2sqrt{2n}}{2} = frac{1}{2} + sqrt{2n} ]Yes! That works. So, combining these steps, I've shown that ( k ) must be less than ( frac{1}{2} + sqrt{2n} ).I think that makes sense. By counting the number of triplets and using the given conditions, I derived the required inequality. It's a bit involved, but I think each step follows logically.

💡Okay, so I have this problem here where I need to find the value of (frac{sin 2x}{sin 2y} + frac{cos 2x}{cos 2y}) given that (frac{sin x}{sin y} = 4) and (frac{cos x}{cos y} = frac{1}{3}). Hmm, let me try to figure this out step by step.First, I remember that there are double-angle identities for sine and cosine. For sine, it's (sin 2theta = 2sinthetacostheta), and for cosine, there are a few forms, but one common one is (cos 2theta = 1 - 2sin^2theta). Maybe I can use these identities to express (sin 2x) and (cos 2x) in terms of (sin x), (cos x), (sin y), and (cos y).Starting with (frac{sin 2x}{sin 2y}), using the double-angle identity for sine, this becomes:[frac{sin 2x}{sin 2y} = frac{2sin x cos x}{2sin y cos y}]Oh, the 2's cancel out, so it simplifies to:[frac{sin x}{sin y} cdot frac{cos x}{cos y}]We already know (frac{sin x}{sin y} = 4) and (frac{cos x}{cos y} = frac{1}{3}), so plugging those in:[4 times frac{1}{3} = frac{4}{3}]Okay, so that part is straightforward. Now, moving on to (frac{cos 2x}{cos 2y}). Using the identity (cos 2theta = 1 - 2sin^2theta), we can write:[frac{cos 2x}{cos 2y} = frac{1 - 2sin^2 x}{1 - 2sin^2 y}]Hmm, but I don't know (sin^2 x) or (sin^2 y) directly. However, I do know the ratios of (sin x) to (sin y) and (cos x) to (cos y). Maybe I can express (sin x) and (cos x) in terms of (sin y) and (cos y), and then use the Pythagorean identity to find (sin^2 y).Given (frac{sin x}{sin y} = 4), that means (sin x = 4sin y). Similarly, (frac{cos x}{cos y} = frac{1}{3}) implies (cos x = frac{1}{3}cos y). Now, using the Pythagorean identity for (x):[sin^2 x + cos^2 x = 1]Substituting the expressions in terms of (y):[(4sin y)^2 + left(frac{1}{3}cos yright)^2 = 1]Calculating each term:[16sin^2 y + frac{1}{9}cos^2 y = 1]But I also know that (cos^2 y = 1 - sin^2 y), so substituting that in:[16sin^2 y + frac{1}{9}(1 - sin^2 y) = 1]Let me simplify this equation:[16sin^2 y + frac{1}{9} - frac{1}{9}sin^2 y = 1]Combine like terms:[left(16 - frac{1}{9}right)sin^2 y + frac{1}{9} = 1]Calculating (16 - frac{1}{9}):[frac{144}{9} - frac{1}{9} = frac{143}{9}]So the equation becomes:[frac{143}{9}sin^2 y + frac{1}{9} = 1]Subtract (frac{1}{9}) from both sides:[frac{143}{9}sin^2 y = frac{8}{9}]Multiply both sides by (frac{9}{143}):[sin^2 y = frac{8}{143}]Wait, that seems a bit off. Let me double-check my calculations. Starting from:[16sin^2 y + frac{1}{9}cos^2 y = 1]Substituting (cos^2 y = 1 - sin^2 y):[16sin^2 y + frac{1}{9}(1 - sin^2 y) = 1]Expanding:[16sin^2 y + frac{1}{9} - frac{1}{9}sin^2 y = 1]Combine the (sin^2 y) terms:[left(16 - frac{1}{9}right)sin^2 y + frac{1}{9} = 1]Calculating (16 - frac{1}{9}):[frac{144}{9} - frac{1}{9} = frac{143}{9}]So:[frac{143}{9}sin^2 y + frac{1}{9} = 1]Subtract (frac{1}{9}):[frac{143}{9}sin^2 y = frac{8}{9}]Multiply both sides by (frac{9}{143}):[sin^2 y = frac{8}{143}]Wait, that seems correct. So (sin^2 y = frac{8}{143}). Therefore, (sin y = sqrt{frac{8}{143}}) or (-sqrt{frac{8}{143}}), but since we're dealing with squares, it doesn't matter for now.Now, let's find (sin^2 x). Since (sin x = 4sin y), squaring both sides:[sin^2 x = 16sin^2 y = 16 times frac{8}{143} = frac{128}{143}]Similarly, (cos^2 x = left(frac{1}{3}cos yright)^2 = frac{1}{9}cos^2 y). But (cos^2 y = 1 - sin^2 y = 1 - frac{8}{143} = frac{135}{143}). So:[cos^2 x = frac{1}{9} times frac{135}{143} = frac{15}{143}]Wait, let me check that calculation again. (cos^2 y = 1 - frac{8}{143} = frac{135}{143}). Then (cos^2 x = frac{1}{9} times frac{135}{143}). Calculating (frac{135}{9} = 15), so (cos^2 x = frac{15}{143}). That seems correct.Now, going back to (cos 2x) and (cos 2y). Using the identity (cos 2theta = 1 - 2sin^2theta):For (cos 2x):[cos 2x = 1 - 2sin^2 x = 1 - 2 times frac{128}{143} = 1 - frac{256}{143}]Calculating (1) as (frac{143}{143}):[frac{143}{143} - frac{256}{143} = -frac{113}{143}]Wait, that can't be right because (frac{256}{143}) is greater than 1, so (cos 2x) would be negative, which is possible, but let me verify the calculation.Wait, (sin^2 x = frac{128}{143}), so (2sin^2 x = frac{256}{143}). Then (1 - frac{256}{143}) is indeed (-frac{113}{143}). Okay, that seems correct.For (cos 2y):[cos 2y = 1 - 2sin^2 y = 1 - 2 times frac{8}{143} = 1 - frac{16}{143}]Again, (1 = frac{143}{143}), so:[frac{143}{143} - frac{16}{143} = frac{127}{143}]So, (cos 2y = frac{127}{143}).Now, (frac{cos 2x}{cos 2y} = frac{-frac{113}{143}}{frac{127}{143}} = -frac{113}{127}).Wait, but earlier I thought (sin^2 y = frac{8}{145}), but now I have (sin^2 y = frac{8}{143}). Did I make a mistake earlier?Let me go back to the step where I substituted (cos^2 y = 1 - sin^2 y) into the equation:[16sin^2 y + frac{1}{9}(1 - sin^2 y) = 1]Expanding:[16sin^2 y + frac{1}{9} - frac{1}{9}sin^2 y = 1]Combine terms:[left(16 - frac{1}{9}right)sin^2 y + frac{1}{9} = 1]Calculating (16 - frac{1}{9}):[frac{144}{9} - frac{1}{9} = frac{143}{9}]So:[frac{143}{9}sin^2 y + frac{1}{9} = 1]Subtract (frac{1}{9}):[frac{143}{9}sin^2 y = frac{8}{9}]Multiply both sides by (frac{9}{143}):[sin^2 y = frac{8}{143}]Yes, that's correct. So (sin^2 y = frac{8}{143}), not (frac{8}{145}). I must have made a typo earlier. So, (cos 2y = frac{127}{143}) and (cos 2x = -frac{113}{143}).Therefore, (frac{cos 2x}{cos 2y} = -frac{113}{127}).Wait, but let me check the calculation for (cos 2x) again. (sin^2 x = frac{128}{143}), so (2sin^2 x = frac{256}{143}). Then (1 - frac{256}{143} = -frac{113}{143}). That's correct.Similarly, (sin^2 y = frac{8}{143}), so (2sin^2 y = frac{16}{143}). Then (1 - frac{16}{143} = frac{127}{143}). Correct.So, (frac{cos 2x}{cos 2y} = -frac{113}{127}).Now, going back to the original expression:[frac{sin 2x}{sin 2y} + frac{cos 2x}{cos 2y} = frac{4}{3} + left(-frac{113}{127}right)]To add these fractions, I need a common denominator. The denominators are 3 and 127. The least common multiple of 3 and 127 is 381.Converting (frac{4}{3}) to have denominator 381:[frac{4}{3} = frac{4 times 127}{3 times 127} = frac{508}{381}]Converting (-frac{113}{127}) to have denominator 381:[-frac{113}{127} = -frac{113 times 3}{127 times 3} = -frac{339}{381}]Now, adding them together:[frac{508}{381} - frac{339}{381} = frac{508 - 339}{381} = frac{169}{381}]Wait, 508 - 339 is 169? Let me check: 508 - 300 = 208, then subtract 39 more: 208 - 39 = 169. Yes, that's correct.So, the sum is (frac{169}{381}). Can this fraction be simplified? Let's see if 169 and 381 have any common factors.169 is 13 squared, so 13 is a factor. Let's check if 13 divides 381. 13 times 29 is 377, and 381 - 377 = 4, so no, 13 doesn't divide 381. Therefore, (frac{169}{381}) is in its simplest form.Wait, but earlier I thought the answer was (frac{61}{129}). Did I make a mistake somewhere? Let me go back through my steps.Wait, in the initial problem, the user wrote the final answer as (boxed{frac{61}{129}}), but in my calculation, I ended up with (frac{169}{381}). Hmm, that suggests I might have made an error in my calculations.Let me double-check the step where I calculated (sin^2 y). Starting from:[16sin^2 y + frac{1}{9}cos^2 y = 1]Substituting (cos^2 y = 1 - sin^2 y):[16sin^2 y + frac{1}{9}(1 - sin^2 y) = 1]Expanding:[16sin^2 y + frac{1}{9} - frac{1}{9}sin^2 y = 1]Combining like terms:[left(16 - frac{1}{9}right)sin^2 y + frac{1}{9} = 1]Calculating (16 - frac{1}{9}):[frac{144}{9} - frac{1}{9} = frac{143}{9}]So:[frac{143}{9}sin^2 y + frac{1}{9} = 1]Subtracting (frac{1}{9}):[frac{143}{9}sin^2 y = frac{8}{9}]Multiplying both sides by (frac{9}{143}):[sin^2 y = frac{8}{143}]That seems correct. So, (sin^2 y = frac{8}{143}).Then, (sin^2 x = 16sin^2 y = frac{128}{143}), and (cos^2 x = frac{1}{9}cos^2 y = frac{1}{9}(1 - frac{8}{143}) = frac{1}{9} times frac{135}{143} = frac{15}{143}).Using these, (cos 2x = 1 - 2sin^2 x = 1 - frac{256}{143} = -frac{113}{143}), and (cos 2y = 1 - 2sin^2 y = 1 - frac{16}{143} = frac{127}{143}).Thus, (frac{cos 2x}{cos 2y} = -frac{113}{127}).Adding to (frac{4}{3}):[frac{4}{3} - frac{113}{127}]Finding a common denominator, which is 381:[frac{4 times 127}{3 times 127} - frac{113 times 3}{127 times 3} = frac{508}{381} - frac{339}{381} = frac{169}{381}]Simplifying (frac{169}{381}): 169 is 13², and 381 divided by 13 is 29.307, which is not an integer, so it can't be simplified further.Wait, but the initial answer was (frac{61}{129}). Did I make a mistake in the calculation of (cos 2x) or (cos 2y)?Let me check (cos 2x) again. (cos 2x = 1 - 2sin^2 x = 1 - 2 times frac{128}{143} = 1 - frac{256}{143}). Since (frac{256}{143} = 1.7902), so (1 - 1.7902 = -0.7902), which is approximately (-frac{113}{143}) because (frac{113}{143} approx 0.7902). So that's correct.Similarly, (cos 2y = 1 - 2sin^2 y = 1 - 2 times frac{8}{143} = 1 - frac{16}{143} = frac{127}{143}), which is approximately 0.888. So that's correct.Therefore, (frac{cos 2x}{cos 2y} = -frac{113}{127}).Adding to (frac{4}{3}), which is approximately 1.333, and (-frac{113}{127} approx -0.889), so the sum is approximately 1.333 - 0.889 = 0.444, which is approximately (frac{169}{381} approx 0.443). But the initial answer was (frac{61}{129} approx 0.472), which is slightly different. Hmm, perhaps I made a mistake in the calculation of (sin^2 y).Wait, let me go back to the step where I substituted (cos^2 y = 1 - sin^2 y) into the equation:[16sin^2 y + frac{1}{9}cos^2 y = 1]Which becomes:[16sin^2 y + frac{1}{9}(1 - sin^2 y) = 1]Expanding:[16sin^2 y + frac{1}{9} - frac{1}{9}sin^2 y = 1]Combining like terms:[left(16 - frac{1}{9}right)sin^2 y + frac{1}{9} = 1]Calculating (16 - frac{1}{9}):[frac{144}{9} - frac{1}{9} = frac{143}{9}]So:[frac{143}{9}sin^2 y + frac{1}{9} = 1]Subtracting (frac{1}{9}):[frac{143}{9}sin^2 y = frac{8}{9}]Multiplying both sides by (frac{9}{143}):[sin^2 y = frac{8}{143}]Wait, that's correct. So, perhaps the initial answer was incorrect, and the correct answer is (frac{169}{381}). Alternatively, maybe I made a mistake in the initial step.Wait, let me check the initial problem again. It says (frac{sin x}{sin y} = 4) and (frac{cos x}{cos y} = frac{1}{3}). So, (sin x = 4sin y) and (cos x = frac{1}{3}cos y).Using the Pythagorean identity:[sin^2 x + cos^2 x = 1]Substituting:[(4sin y)^2 + left(frac{1}{3}cos yright)^2 = 1]Which is:[16sin^2 y + frac{1}{9}cos^2 y = 1]Substituting (cos^2 y = 1 - sin^2 y):[16sin^2 y + frac{1}{9}(1 - sin^2 y) = 1]Which simplifies to:[frac{143}{9}sin^2 y + frac{1}{9} = 1]So, (sin^2 y = frac{8}{143}). That seems correct.Therefore, (sin^2 x = 16 times frac{8}{143} = frac{128}{143}), and (cos^2 x = frac{1}{9} times frac{135}{143} = frac{15}{143}).Thus, (cos 2x = 1 - 2 times frac{128}{143} = -frac{113}{143}), and (cos 2y = 1 - 2 times frac{8}{143} = frac{127}{143}).So, (frac{cos 2x}{cos 2y} = -frac{113}{127}).Adding to (frac{sin 2x}{sin 2y} = frac{4}{3}), we get:[frac{4}{3} - frac{113}{127} = frac{508}{381} - frac{339}{381} = frac{169}{381}]Simplifying (frac{169}{381}), since 169 is 13² and 381 is 3 × 127, which are both primes, so it can't be simplified further.Wait, but the initial answer was (frac{61}{129}). Let me check if (frac{169}{381}) simplifies to (frac{61}{129}). Calculating (frac{169}{381}) divided by (frac{61}{129}):[frac{169}{381} div frac{61}{129} = frac{169}{381} times frac{129}{61} = frac{169 times 129}{381 times 61}]Simplifying, 381 ÷ 129 = 3, so:[frac{169 times 129}{381 times 61} = frac{169 times 1}{3 times 61} = frac{169}{183}]Which is not equal to 1, so they are different fractions. Therefore, my calculation must be correct, and the initial answer might have been incorrect.Alternatively, perhaps I made a mistake in the calculation of (cos 2x) or (cos 2y). Let me double-check.Using (cos 2theta = 1 - 2sin^2theta):For (cos 2x):[cos 2x = 1 - 2sin^2 x = 1 - 2 times frac{128}{143} = 1 - frac{256}{143}]Calculating (1 = frac{143}{143}), so:[frac{143}{143} - frac{256}{143} = -frac{113}{143}]Correct.For (cos 2y):[cos 2y = 1 - 2sin^2 y = 1 - 2 times frac{8}{143} = 1 - frac{16}{143} = frac{127}{143}]Correct.Therefore, (frac{cos 2x}{cos 2y} = -frac{113}{127}).Adding to (frac{4}{3}), we get:[frac{4}{3} - frac{113}{127} = frac{508}{381} - frac{339}{381} = frac{169}{381}]So, the correct answer should be (frac{169}{381}), which simplifies to (frac{169}{381}) since it can't be reduced further.Wait, but (frac{169}{381}) can be simplified by dividing numerator and denominator by GCD(169,381). Let's find the GCD.Prime factors of 169: 13 × 13.Prime factors of 381: 3 × 127.No common factors, so (frac{169}{381}) is in simplest form.Therefore, the final answer is (frac{169}{381}).But the initial answer was (frac{61}{129}), which is approximately 0.472, while (frac{169}{381}) is approximately 0.443. These are close but not the same. It seems there was a mistake in the initial calculation, possibly in the step where (sin^2 y) was calculated as (frac{8}{145}) instead of (frac{8}{143}).Therefore, the correct answer should be (frac{169}{381}), which can be simplified by dividing numerator and denominator by GCD(169,381). Since GCD is 1, it remains (frac{169}{381}).Alternatively, perhaps I made a mistake in the initial step of calculating (sin^2 y). Let me check again.Starting from:[16sin^2 y + frac{1}{9}cos^2 y = 1]Substituting (cos^2 y = 1 - sin^2 y):[16sin^2 y + frac{1}{9}(1 - sin^2 y) = 1]Expanding:[16sin^2 y + frac{1}{9} - frac{1}{9}sin^2 y = 1]Combining like terms:[left(16 - frac{1}{9}right)sin^2 y + frac{1}{9} = 1]Calculating (16 - frac{1}{9}):[frac{144}{9} - frac{1}{9} = frac{143}{9}]So:[frac{143}{9}sin^2 y + frac{1}{9} = 1]Subtracting (frac{1}{9}):[frac{143}{9}sin^2 y = frac{8}{9}]Multiplying both sides by (frac{9}{143}):[sin^2 y = frac{8}{143}]Yes, that's correct. So, the initial answer was incorrect, and the correct answer is (frac{169}{381}).Alternatively, perhaps the initial problem had a typo, and the ratios were different. But assuming the problem is correct as stated, the answer should be (frac{169}{381}).Wait, but (frac{169}{381}) can be simplified by dividing numerator and denominator by 13:169 ÷ 13 = 13381 ÷ 13 = 29.307, which is not an integer. So, it can't be simplified further.Therefore, the final answer is (frac{169}{381}).But let me check if (frac{169}{381}) can be simplified by dividing numerator and denominator by 13:169 ÷ 13 = 13381 ÷ 13 = 29.307, which is not an integer. So, it can't be simplified further.Therefore, the correct answer is (frac{169}{381}).Wait, but 169 is 13², and 381 is 3 × 127. Since 127 is a prime number, and 13 doesn't divide 127, the fraction can't be simplified further.Therefore, the final answer is (frac{169}{381}).But wait, let me check if I made a mistake in the calculation of (cos 2x) and (cos 2y). Maybe I should use a different identity for (cos 2theta), such as (cos 2theta = 2cos^2theta - 1), to see if I get the same result.Using (cos 2theta = 2cos^2theta - 1):For (cos 2x):[cos 2x = 2cos^2 x - 1 = 2 times frac{15}{143} - 1 = frac{30}{143} - 1 = frac{30 - 143}{143} = -frac{113}{143}]Same result.For (cos 2y):[cos 2y = 2cos^2 y - 1 = 2 times frac{135}{143} - 1 = frac{270}{143} - 1 = frac{270 - 143}{143} = frac{127}{143}]Same result.Therefore, (frac{cos 2x}{cos 2y} = -frac{113}{127}), which confirms my earlier calculation.Thus, the final answer is (frac{4}{3} - frac{113}{127} = frac{169}{381}).I think that's the correct answer. The initial answer might have had a typo or calculation error.

💡Alright, so I've got this problem here about finding the median of a set of 100 numbers. Let me try to wrap my head around it step by step. First off, the problem says that if one number is removed from the set, the median of the remaining 99 numbers is 78. Then, if another number is removed, making it 98 numbers, the median drops to 66. I need to find the median of the original set of 100 numbers. Hmm, okay.Let me recall what a median is. The median is the middle value in an ordered list of numbers. If there's an even number of observations, the median is the average of the two middle numbers. So, for 100 numbers, the median would be the average of the 50th and 51st numbers when they're sorted in order.Now, when we remove one number, we're left with 99 numbers. Since 99 is odd, the median will be the 50th number in this new list. The problem states that this median is 78. So, the 50th number in the list of 99 numbers is 78.Wait, but how does removing a number affect the position of the median? Let me think. If I remove a number from the original set of 100, depending on where that number is, it could shift the median. If I remove a number that's less than or equal to the original median, the median might stay the same or increase. If I remove a number that's greater than or equal to the original median, the median might decrease or stay the same.But in this case, after removing one number, the median becomes 78. So, the original median must have been influenced by this removal. Let me denote the original 100 numbers in ascending order as ( a_1, a_2, a_3, ldots, a_{100} ). The original median would be the average of ( a_{50} ) and ( a_{51} ).After removing one number, say ( a_k ), the new list has 99 numbers. The median of this new list is the 50th number, which is given as 78. So, ( a_{50} ) in the new list is 78. But this ( a_{50} ) corresponds to either ( a_{50} ) or ( a_{51} ) in the original list, depending on whether ( k ) was less than or equal to 50 or greater than 50.Wait, if we remove a number from the first 50 numbers, then the 51st number in the original list becomes the 50th number in the new list. Similarly, if we remove a number from the last 50 numbers, the 50th number in the original list remains the 50th number in the new list.But the median after removal is 78, which is higher than the original median, which we don't know yet. Hmm, maybe I need to consider both scenarios.Case 1: Removing a number from the first 50 numbers. Then, the 51st number becomes the 50th number in the new list, which is 78. So, ( a_{51} = 78 ).Case 2: Removing a number from the last 50 numbers. Then, the 50th number remains the 50th number in the new list, which is 78. So, ( a_{50} = 78 ).But wait, the problem doesn't specify where the number was removed from. It just says one number is removed, and the median becomes 78. So, both cases are possible. Hmm, but then when another number is removed, making it 98 numbers, the median drops to 66.Let me think about that second removal. Now, with 98 numbers, the median is the average of the 49th and 50th numbers. The problem says this median is 66. So, the average of the 49th and 50th numbers is 66. That means both ( a_{49} ) and ( a_{50} ) are 66, or one is 66 and the other is 66 as well? Wait, no, the average is 66, so the sum of the 49th and 50th numbers is 132.But hold on, in the second removal, we're removing another number from the 99-number list. Depending on where we remove this number, the median could shift again. If we remove a number from the first 49 numbers, the 50th number becomes the 49th, and the new median would be the average of the 49th and 50th numbers, which would be the original 50th and 51st numbers. But since the median is 66, that suggests that both ( a_{50} ) and ( a_{51} ) are 66? Wait, but earlier, in the first removal, we had either ( a_{50} = 78 ) or ( a_{51} = 78 ).This is getting a bit confusing. Let me try to structure this.Original set: 100 numbers, median is average of ( a_{50} ) and ( a_{51} ).After removing one number, 99 numbers left. Median is 78, which is ( a_{50} ) in the new list.After removing another number, 98 numbers left. Median is 66, which is the average of ( a_{49} ) and ( a_{50} ) in the new list.Wait, no. For 98 numbers, the median is the average of the 49th and 50th numbers.But in the second removal, we're removing a number from the 99-number list. So, depending on where we remove it, the median could shift.Let me consider the first removal:If we remove a number from the first 50, then the new list's 50th number is the original 51st number, which is 78. So, ( a_{51} = 78 ).Then, in the second removal, if we remove a number from the last 50 (positions 51 to 100 in the original list), the new list's 49th and 50th numbers would be the original 49th and 50th numbers. But the median is 66, so the average of these two is 66. Therefore, ( a_{49} + a_{50} = 132 ).But from the first removal, we have ( a_{51} = 78 ). So, the original median was the average of ( a_{50} ) and 78. If ( a_{49} + a_{50} = 132 ), then ( a_{50} ) could be 66, making ( a_{49} = 66 ) as well. So, the original median would be ( (66 + 78)/2 = 72 ).Alternatively, if in the first removal, we removed a number from the last 50, making ( a_{50} = 78 ). Then, in the second removal, if we remove a number from the first 50, the new list's 49th and 50th numbers would be the original 50th and 51st numbers. But the median is 66, so the average of these two is 66. Therefore, ( a_{50} + a_{51} = 132 ). But since ( a_{50} = 78 ), this would imply ( a_{51} = 54 ), which doesn't make sense because the list is sorted in ascending order, so ( a_{51} ) should be greater than or equal to ( a_{50} ).Therefore, this scenario is impossible because ( a_{51} ) can't be less than ( a_{50} ). So, the first removal must have been from the first 50 numbers, making ( a_{51} = 78 ), and the second removal from the last 50 numbers, making ( a_{49} + a_{50} = 132 ). Since the list is sorted, ( a_{49} leq a_{50} leq a_{51} ). Given that ( a_{51} = 78 ), ( a_{50} ) must be less than or equal to 78. If ( a_{49} + a_{50} = 132 ), and ( a_{50} leq 78 ), then ( a_{49} geq 54 ). But since ( a_{49} leq a_{50} ), the only way for their sum to be 132 is if both are 66. Therefore, ( a_{49} = a_{50} = 66 ).Thus, the original median is the average of ( a_{50} = 66 ) and ( a_{51} = 78 ), which is ( (66 + 78)/2 = 72 ).Wait, let me double-check. If ( a_{50} = 66 ) and ( a_{51} = 78 ), then the original median is indeed 72. After removing a number from the first 50, say ( a_{50} ), the new list's 50th number is ( a_{51} = 78 ). Then, removing a number from the last 50, say ( a_{51} ), the new list's 49th and 50th numbers are ( a_{49} = 66 ) and ( a_{50} = 66 ), whose average is 66. That fits the problem's description.Alternatively, if we removed ( a_{51} ) first, making the new median ( a_{50} = 78 ), but then removing another number from the last 50 would require ( a_{49} + a_{50} = 132 ), which would imply ( a_{49} = 54 ) if ( a_{50} = 78 ), but that's not possible because ( a_{49} leq a_{50} ). So, the only valid scenario is removing from the first 50 first.Therefore, the original median must be 72.

💡Okay, so I have this geometry problem here about triangle DEF. The sides are DE = 14, DF = 15, and EF = 21. The medians DM, EN, and FO intersect at the centroid G. I need to find the length of GQ, where Q is the foot of the altitude from G to EF.Hmm, let me start by recalling what a centroid is. The centroid of a triangle is the point where all three medians intersect, and it divides each median into a ratio of 2:1, with the longer part being closer to the vertex. So, if I can find the length of the median from D to EF, then I can find the distance from G to EF by taking one-third of that median's length.Wait, but actually, GQ is the altitude from G to EF, not necessarily along the median. Hmm, so maybe I need to find the height from G to EF, not just along the median. That might be a bit different.Let me think. Maybe I can find the area of the triangle first, and then use that to find the height from D to EF. Since the centroid divides the median into a 2:1 ratio, the height from G to EF would be one-third of the height from D to EF.Okay, that sounds plausible. So, first, I need to find the area of triangle DEF. I can use Heron's formula for that. Heron's formula says that the area of a triangle with sides a, b, c is sqrt[s(s - a)(s - b)(s - c)], where s is the semi-perimeter.So, let's compute the semi-perimeter, s, of triangle DEF. The sides are 14, 15, and 21. So, s = (14 + 15 + 21)/2 = (50)/2 = 25.Now, using Heron's formula, the area A is sqrt[s(s - DE)(s - DF)(s - EF)] = sqrt[25(25 - 14)(25 - 15)(25 - 21)] = sqrt[25 * 11 * 10 * 4].Let me compute that step by step. 25 * 11 is 275, 275 * 10 is 2750, 2750 * 4 is 11000. So, sqrt[11000]. Hmm, sqrt[11000] can be simplified. 11000 is 100 * 110, so sqrt[100 * 110] = 10 * sqrt[110]. Wait, but 110 can be broken down further. 110 is 10 * 11, so sqrt[110] is sqrt[10*11], which doesn't simplify further. So, the area is 10*sqrt[110]. Wait, but 11000 is 100*110, so sqrt[100*110] = 10*sqrt[110]. Hmm, but 110 is 10*11, so sqrt[10*11] is just sqrt[110]. So, the area is 10*sqrt[110].Wait, but let me double-check that. 25*11 is 275, 275*10 is 2750, 2750*4 is 11000. So, sqrt[11000]. Let me see, 11000 is 100*110, so sqrt[100*110] is 10*sqrt[110]. So, yes, that's correct.But wait, 110 can be written as 10*11, so sqrt[110] is sqrt[10*11], which is sqrt[10]*sqrt[11]. So, 10*sqrt[110] is 10*sqrt[10]*sqrt[11]. Hmm, but maybe I can write it as 10*sqrt[110]. Either way, it's fine.Now, the area is 10*sqrt[110]. Alternatively, I can write it as 10*sqrt[110] or 10*sqrt[10*11]. Either way, it's approximately, but I don't need the approximate value.Now, I need to find the height from D to EF. Let's denote this height as h. The area of the triangle can also be expressed as (1/2)*base*height. Here, the base is EF, which is 21, and the height is h.So, area = (1/2)*21*h = (21/2)*h.But we already found the area as 10*sqrt[110]. So, setting them equal:(21/2)*h = 10*sqrt[110]Solving for h:h = (10*sqrt[110]) / (21/2) = (10*sqrt[110]) * (2/21) = (20*sqrt[110])/21.So, the height from D to EF is (20*sqrt[110])/21.Now, since G is the centroid, it divides the median into a ratio of 2:1. So, the distance from G to EF would be one-third of the height from D to EF.Wait, is that correct? Because the centroid is located at one-third of the median from the base. So, if the height from D to EF is h, then the distance from G to EF would be h/3.So, GQ = h/3 = [(20*sqrt[110])/21]/3 = (20*sqrt[110])/63.Simplifying, 20 and 63 have a common factor of... 20 is 4*5, 63 is 7*9, so no common factors. So, GQ = (20*sqrt[110])/63.Wait, but let me make sure. Is the height from G to EF equal to one-third of the height from D to EF?Yes, because the centroid divides the median into a 2:1 ratio, so the distance from the centroid to the base is one-third of the height from the vertex to the base.So, that seems correct.But let me double-check the calculations.First, semi-perimeter s = (14 + 15 + 21)/2 = 50/2 = 25. Correct.Area = sqrt[25*(25 - 14)*(25 - 15)*(25 - 21)] = sqrt[25*11*10*4] = sqrt[11000] = 10*sqrt[110]. Correct.Height from D to EF: area = (1/2)*21*h => h = (2*area)/21 = (2*10*sqrt[110])/21 = (20*sqrt[110])/21. Correct.Then, GQ = h/3 = (20*sqrt[110])/63. Correct.Wait, but in the initial problem statement, the sides are DE = 14, DF = 15, EF = 21. So, triangle DEF with sides 14, 15, 21.Wait, but in my calculation, I used DE = 14, DF = 15, EF = 21, so that's correct.Alternatively, maybe I can use coordinates to verify.Let me try to assign coordinates to the triangle to compute GQ.Let me place point E at (0, 0), point F at (21, 0), and point D somewhere in the plane.Then, the coordinates of D can be found using the distances DE = 14 and DF = 15.So, coordinates of E = (0, 0), F = (21, 0). Let D = (x, y).Then, distance from D to E is 14: sqrt[(x - 0)^2 + (y - 0)^2] = 14 => x^2 + y^2 = 196.Distance from D to F is 15: sqrt[(x - 21)^2 + (y - 0)^2] = 15 => (x - 21)^2 + y^2 = 225.Subtracting the first equation from the second:(x - 21)^2 + y^2 - (x^2 + y^2) = 225 - 196 => x^2 - 42x + 441 + y^2 - x^2 - y^2 = 29 => -42x + 441 = 29 => -42x = 29 - 441 = -412 => x = (-412)/(-42) = 412/42.Simplify 412/42: divide numerator and denominator by 2: 206/21. So, x = 206/21.Now, plug x back into x^2 + y^2 = 196:(206/21)^2 + y^2 = 196.Compute (206)^2: 206*206. Let's compute 200^2 = 40000, 6^2=36, and 2*200*6=2400. So, (200 + 6)^2 = 40000 + 2400 + 36 = 42436.So, (206/21)^2 = 42436 / 441.Thus, y^2 = 196 - (42436 / 441).Convert 196 to a fraction over 441: 196 = 196 * (441/441) = (196*441)/441.Compute 196*441: 196*400=78400, 196*41=8036, so total is 78400 + 8036 = 86436.So, y^2 = (86436 - 42436)/441 = (44000)/441.Thus, y = sqrt(44000/441) = sqrt(44000)/21.Simplify sqrt(44000): 44000 = 100*440, so sqrt(100*440) = 10*sqrt(440).Simplify sqrt(440): 440 = 4*110, so sqrt(440) = 2*sqrt(110).Thus, y = 10*2*sqrt(110)/21 = 20*sqrt(110)/21.So, coordinates of D are (206/21, 20*sqrt(110)/21).Now, the centroid G is the average of the coordinates of D, E, and F.Coordinates of E = (0, 0), F = (21, 0), D = (206/21, 20*sqrt(110)/21).So, centroid G has coordinates:x_G = (0 + 21 + 206/21)/3, y_G = (0 + 0 + 20*sqrt(110)/21)/3.Compute x_G:First, convert 21 to 21/1, so 21 = 441/21.Thus, x_G = (0 + 441/21 + 206/21)/3 = (441 + 206)/21 /3 = (647/21)/3 = 647/(21*3) = 647/63.Similarly, y_G = (20*sqrt(110)/21)/3 = (20*sqrt(110))/(21*3) = (20*sqrt(110))/63.So, centroid G is at (647/63, 20*sqrt(110)/63).Now, we need to find the foot of the altitude from G to EF. Since EF is along the x-axis from (0,0) to (21,0), the foot Q will have the same x-coordinate as G, but y-coordinate 0. Wait, no, that's only if the altitude is vertical, but in this case, EF is horizontal, so the altitude from G to EF is vertical, so yes, Q would be (x_G, 0).Wait, is that correct? If EF is along the x-axis, then the altitude from any point to EF is just the vertical distance to the x-axis, so yes, Q would be (x_G, 0).But wait, actually, no. The foot of the altitude from G to EF is the perpendicular projection of G onto EF. Since EF is horizontal, the perpendicular is vertical, so yes, Q is (x_G, 0).Therefore, the length GQ is the vertical distance from G to Q, which is just the y-coordinate of G, which is 20*sqrt(110)/63.So, GQ = 20*sqrt(110)/63.Wait, but earlier I thought it was 100*sqrt(22)/63. Hmm, that's different. Did I make a mistake somewhere?Wait, let me check. In the initial approach, I found the height from D to EF as (20*sqrt(110))/21, then took one-third of that to get GQ as (20*sqrt(110))/63.In the coordinate approach, I found the y-coordinate of G as (20*sqrt(110))/63, which is the same as GQ because Q is (x_G, 0). So, both methods give the same result.Wait, but in the initial problem statement, the answer was given as 100*sqrt(22)/63. So, which one is correct?Wait, let me check my Heron's formula calculation again.s = 25, area = sqrt[25*11*10*4] = sqrt[11000]. Now, 11000 = 100*110, so sqrt[11000] = 10*sqrt[110]. So, area is 10*sqrt[110).Then, height from D to EF: area = (1/2)*21*h => h = (2*10*sqrt(110))/21 = (20*sqrt(110))/21.Then, GQ = h/3 = (20*sqrt(110))/63.But in the initial problem, the answer was 100*sqrt(22)/63. So, perhaps I made a mistake in simplifying sqrt(110).Wait, sqrt(110) is sqrt(11*10), which is sqrt(11)*sqrt(10). Alternatively, 110 is 10*11, so sqrt(110) is just sqrt(110). So, 20*sqrt(110)/63 is correct.But wait, 110 is 10*11, so sqrt(110) is sqrt(10*11). Alternatively, 110 is 2*5*11, so no square factors. So, sqrt(110) is as simplified as it gets.Wait, but in the initial problem, the answer was 100*sqrt(22)/63. So, perhaps I made a mistake in the Heron's formula calculation.Wait, let me recalculate Heron's formula:s = 25.Area = sqrt[s(s - a)(s - b)(s - c)] = sqrt[25*(25 - 14)*(25 - 15)*(25 - 21)] = sqrt[25*11*10*4].25*11 = 275, 275*10 = 2750, 2750*4 = 11000.So, sqrt[11000] = sqrt[100*110] = 10*sqrt[110]. So, that's correct.Wait, but maybe I made a mistake in the coordinate approach. Let me check that again.Coordinates of D: (206/21, 20*sqrt(110)/21).Centroid G: average of D, E, F.E = (0,0), F = (21,0), D = (206/21, 20*sqrt(110)/21).So, x_G = (0 + 21 + 206/21)/3.Convert 21 to 441/21, so x_G = (0 + 441/21 + 206/21)/3 = (647/21)/3 = 647/63.Similarly, y_G = (0 + 0 + 20*sqrt(110)/21)/3 = (20*sqrt(110)/21)/3 = 20*sqrt(110)/63.So, G is at (647/63, 20*sqrt(110)/63).Thus, the foot Q is (647/63, 0), so GQ is the vertical distance, which is 20*sqrt(110)/63.So, both methods give the same result, which is 20*sqrt(110)/63.But in the initial problem, the answer was 100*sqrt(22)/63. So, perhaps there was a miscalculation.Wait, let me see: 20*sqrt(110) vs 100*sqrt(22). Are these equivalent?Wait, sqrt(110) is sqrt(10*11), and sqrt(22) is sqrt(2*11). So, 20*sqrt(110) is 20*sqrt(10*11) = 20*sqrt(10)*sqrt(11).100*sqrt(22) is 100*sqrt(2*11) = 100*sqrt(2)*sqrt(11).So, unless sqrt(10) = 5*sqrt(2), which is not true because sqrt(10) is approximately 3.16 and 5*sqrt(2) is approximately 7.07, which is not equal.So, 20*sqrt(110) is not equal to 100*sqrt(22). Therefore, perhaps I made a mistake in the initial problem's answer.Wait, let me check the initial problem again.Wait, the initial problem was:In triangle DEF, DE = 14, DF = 15, and EF = 21. The medians DM, EN, and FO of triangle DEF intersect at the centroid G. Let Q be the foot of the altitude from G to EF. Find the length of GQ.So, according to my calculations, GQ is 20*sqrt(110)/63.But in the initial problem, the answer was given as 100*sqrt(22)/63.Wait, perhaps I made a mistake in the Heron's formula calculation.Wait, let me recalculate Heron's formula.s = (14 + 15 + 21)/2 = 50/2 = 25.Area = sqrt[s(s - a)(s - b)(s - c)] = sqrt[25*(25 - 14)*(25 - 15)*(25 - 21)] = sqrt[25*11*10*4].25*11 = 275, 275*10 = 2750, 2750*4 = 11000.So, sqrt[11000] = 10*sqrt[110]. So, that's correct.Wait, but maybe the problem was using a different approach, considering the centroid divides the median into 2:1, but perhaps the altitude from G is not one-third of the altitude from D, but something else.Wait, no, because the centroid is located at one-third of the median from the base, so the distance from G to the base is one-third of the height from the vertex to the base.Wait, but in the coordinate approach, I found that the y-coordinate of G is 20*sqrt(110)/63, which is exactly one-third of the height from D to EF, which was 20*sqrt(110)/21.So, that seems correct.Alternatively, perhaps the problem was using a different triangle configuration, but I don't think so.Wait, maybe I made a mistake in the coordinate approach.Wait, in the coordinate approach, I placed E at (0,0), F at (21,0), and D at (x,y). Then, I found x = 206/21 and y = 20*sqrt(110)/21.Then, centroid G is at ( (0 + 21 + 206/21)/3, (0 + 0 + 20*sqrt(110)/21)/3 ) = (647/63, 20*sqrt(110)/63).Thus, GQ is the vertical distance from G to EF, which is just the y-coordinate, so 20*sqrt(110)/63.So, that seems correct.Wait, but perhaps the problem was considering the length of the median DM, and then taking one-third of that as GQ, but that's not necessarily the case because GQ is the altitude, not the median.Wait, the median DM is from D to the midpoint of EF. The midpoint of EF is at (10.5, 0), since EF is from (0,0) to (21,0). So, midpoint M is at (10.5, 0).Then, the median DM is from D(206/21, 20*sqrt(110)/21) to M(10.5, 0).Wait, 10.5 is 21/2, which is 10.5.So, the length of DM can be calculated as the distance between D and M.But perhaps that's not necessary, because GQ is the altitude, not along the median.Wait, but in the coordinate approach, GQ is the vertical distance from G to EF, which is just the y-coordinate of G, which is 20*sqrt(110)/63.So, that seems correct.Wait, but in the initial problem, the answer was 100*sqrt(22)/63. So, perhaps I made a mistake in the Heron's formula calculation.Wait, let me check the Heron's formula again.s = 25.Area = sqrt[25*(25 - 14)*(25 - 15)*(25 - 21)] = sqrt[25*11*10*4].25*11 = 275, 275*10 = 2750, 2750*4 = 11000.So, sqrt[11000] = 10*sqrt[110]. So, that's correct.Wait, but 11000 is 100*110, so sqrt[100*110] = 10*sqrt[110]. So, area is 10*sqrt[110).Then, height from D to EF is (2*area)/21 = (20*sqrt(110))/21.Thus, GQ is one-third of that, which is (20*sqrt(110))/63.So, that seems correct.Wait, but perhaps the problem was considering the length of the median DM, and then taking one-third of that as GQ, but that's not correct because GQ is the altitude, not the median.Wait, the median DM is from D to M, the midpoint of EF. The length of DM can be calculated, but GQ is the altitude from G to EF, which is different.Wait, but in the coordinate approach, GQ is the vertical distance from G to EF, which is just the y-coordinate of G, which is 20*sqrt(110)/63.So, that seems correct.Therefore, I think the correct answer is 20*sqrt(110)/63, not 100*sqrt(22)/63.Wait, but let me check if sqrt(110) can be simplified as sqrt(22*5), which is sqrt(22)*sqrt(5). So, 20*sqrt(110) = 20*sqrt(22*5) = 20*sqrt(22)*sqrt(5). So, 20*sqrt(22)*sqrt(5) is not equal to 100*sqrt(22).Wait, unless sqrt(5) is 5, which it's not. So, perhaps the initial problem's answer was incorrect.Alternatively, maybe I made a mistake in the Heron's formula calculation.Wait, let me try to compute the area using another method, perhaps using coordinates.From the coordinates, the area can be calculated using the shoelace formula.Coordinates of E = (0,0), F = (21,0), D = (206/21, 20*sqrt(110)/21).So, shoelace formula:Area = (1/2)| (x_E(y_F - y_D) + x_F(y_D - y_E) + x_D(y_E - y_F)) |.Plugging in:= (1/2)| 0*(0 - 20*sqrt(110)/21) + 21*(20*sqrt(110)/21 - 0) + (206/21)*(0 - 0) |= (1/2)| 0 + 21*(20*sqrt(110)/21) + 0 |= (1/2)| 20*sqrt(110) | = (1/2)*20*sqrt(110) = 10*sqrt(110).So, that's consistent with Heron's formula. So, area is indeed 10*sqrt(110).Thus, height from D to EF is (2*area)/21 = (20*sqrt(110))/21.Then, GQ is one-third of that, which is (20*sqrt(110))/63.So, I think that's correct.Therefore, the length of GQ is 20*sqrt(110)/63.Wait, but in the initial problem, the answer was 100*sqrt(22)/63. So, perhaps there was a miscalculation there.Alternatively, perhaps I made a mistake in the coordinate approach.Wait, let me check the coordinates again.From E(0,0), F(21,0), D(x,y).Distance DE = 14: x^2 + y^2 = 196.Distance DF = 15: (x - 21)^2 + y^2 = 225.Subtracting: (x - 21)^2 + y^2 - x^2 - y^2 = 225 - 196 => x^2 - 42x + 441 + y^2 - x^2 - y^2 = 29 => -42x + 441 = 29 => -42x = -412 => x = 412/42 = 206/21.Then, x = 206/21.Then, x^2 + y^2 = 196.Compute x^2: (206/21)^2 = (206)^2 / (21)^2 = 42436 / 441.Thus, y^2 = 196 - 42436/441.Convert 196 to 196*441/441 = 86436/441.Thus, y^2 = (86436 - 42436)/441 = 44000/441.Thus, y = sqrt(44000)/21 = sqrt(100*440)/21 = 10*sqrt(440)/21.Simplify sqrt(440): sqrt(4*110) = 2*sqrt(110).Thus, y = 10*2*sqrt(110)/21 = 20*sqrt(110)/21.So, that's correct.Thus, coordinates of D are (206/21, 20*sqrt(110)/21).Centroid G is at ( (0 + 21 + 206/21)/3, (0 + 0 + 20*sqrt(110)/21)/3 ).Compute x_G:21 = 441/21, so x_G = (0 + 441/21 + 206/21)/3 = (647/21)/3 = 647/63.Similarly, y_G = (20*sqrt(110)/21)/3 = 20*sqrt(110)/63.Thus, GQ is the vertical distance from G to EF, which is y_G = 20*sqrt(110)/63.So, that's correct.Therefore, the length of GQ is 20*sqrt(110)/63.But wait, the initial problem's answer was 100*sqrt(22)/63. So, perhaps there was a miscalculation in the initial problem.Alternatively, perhaps I made a mistake in interpreting the problem.Wait, the problem says "the foot of the altitude from G to EF." So, in the coordinate system, that's the vertical distance, which is correct.Alternatively, perhaps the problem was considering the length of the median DM, and then taking one-third of that as GQ, but that's not correct because GQ is the altitude, not the median.Wait, the median DM is from D to M, the midpoint of EF. The length of DM can be calculated as the distance between D and M.Coordinates of M: midpoint of EF is (10.5, 0) = (21/2, 0).Coordinates of D: (206/21, 20*sqrt(110)/21).So, distance DM is sqrt[(206/21 - 21/2)^2 + (20*sqrt(110)/21 - 0)^2].Compute 206/21 - 21/2:Convert to common denominator, which is 42.206/21 = (206*2)/42 = 412/42.21/2 = (21*21)/42 = 441/42.Wait, no, 21/2 = (21*21)/42? Wait, no, 21/2 = (21*21)/42 is incorrect.Wait, 21/2 = (21*21)/42? No, that's not correct.Wait, 21/2 = (21*21)/42 is incorrect because 21/2 = (21*21)/42 would imply 21/2 = 441/42, which is 10.5, which is correct because 441/42 = 10.5.Wait, but 206/21 is approximately 9.8095, and 21/2 is 10.5.So, 206/21 - 21/2 = 9.8095 - 10.5 = -0.6905.But let's compute it exactly.206/21 - 21/2 = (206*2 - 21*21)/(21*2) = (412 - 441)/42 = (-29)/42.So, the x-coordinate difference is -29/42.The y-coordinate difference is 20*sqrt(110)/21 - 0 = 20*sqrt(110)/21.Thus, distance DM is sqrt[ (-29/42)^2 + (20*sqrt(110)/21)^2 ].Compute (-29/42)^2 = 841/1764.(20*sqrt(110)/21)^2 = (400*110)/441 = 44000/441.Convert 44000/441 to denominator 1764: 44000/441 = (44000*4)/(441*4) = 176000/1764.Thus, DM^2 = 841/1764 + 176000/1764 = (841 + 176000)/1764 = 176841/1764.Simplify 176841/1764: divide numerator and denominator by 21: 176841 ÷21= 8421, 1764 ÷21=84.So, 8421/84. Wait, 8421 ÷ 84: 84*100=8400, so 8421-8400=21, so 100 + 21/84 = 100 + 1/4 = 100.25.Wait, but 8421/84 = 100.25, which is 401/4.Wait, 401/4 is 100.25, so sqrt(401/4) = sqrt(401)/2 ≈ 20.02499/2 ≈ 10.0125.But that's the length of DM.Then, the centroid G divides DM into a ratio of 2:1, so the distance from G to M is one-third of DM.Thus, GM = DM/3 ≈ 10.0125/3 ≈ 3.3375.But in the coordinate approach, GQ is 20*sqrt(110)/63 ≈ 20*10.488/63 ≈ 209.76/63 ≈ 3.33.So, that's consistent with GM ≈ 3.3375.Wait, but GQ is the altitude from G to EF, which is the vertical distance, which is 20*sqrt(110)/63 ≈ 3.33.So, that's correct.Therefore, the length of GQ is 20*sqrt(110)/63.But in the initial problem, the answer was 100*sqrt(22)/63. So, perhaps the initial problem had a different configuration or a miscalculation.Alternatively, perhaps I made a mistake in simplifying sqrt(110).Wait, sqrt(110) is sqrt(11*10), which is sqrt(11)*sqrt(10). So, 20*sqrt(110) is 20*sqrt(11)*sqrt(10).But 100*sqrt(22) is 100*sqrt(2*11) = 100*sqrt(2)*sqrt(11).So, unless sqrt(10) = 5*sqrt(2), which is not true, these are different.Wait, sqrt(10) is approximately 3.16, and 5*sqrt(2) is approximately 7.07, so they are not equal.Therefore, I think the correct answer is 20*sqrt(110)/63.But to make sure, let me compute both values numerically.Compute 20*sqrt(110)/63:sqrt(110) ≈ 10.488.So, 20*10.488 ≈ 209.76.Divide by 63: 209.76/63 ≈ 3.33.Compute 100*sqrt(22)/63:sqrt(22) ≈ 4.690.So, 100*4.690 ≈ 469.Divide by 63: 469/63 ≈ 7.44.So, 3.33 vs 7.44. These are different.Thus, I think the correct answer is 20*sqrt(110)/63.Therefore, I conclude that the length of GQ is 20*sqrt(110)/63.

💡Okay, so I have this trigonometry problem here. It says that ( csleft(alpha + frac{pi}{2}right) = frac{3}{5} ), and I need to find the value of ( sn2alpha ). The options are given as A: ( frac{12}{25} ), B: ( -frac{12}{25} ), C: ( frac{24}{25} ), and D: ( -frac{24}{25} ). Hmm, I think ( cs ) might be a typo or shorthand for cosine, and ( sn ) might be sine. So, I'll assume ( cs ) is ( cos ) and ( sn ) is ( sin ). That makes more sense.Alright, so the equation is ( cosleft(alpha + frac{pi}{2}right) = frac{3}{5} ). I remember that there are some trigonometric identities that can help simplify expressions like ( cos(theta + phi) ). Specifically, the cosine addition formula: ( cos(A + B) = cos A cos B - sin A sin B ). Let me apply that here.So, ( cosleft(alpha + frac{pi}{2}right) = cosalpha cosleft(frac{pi}{2}right) - sinalpha sinleft(frac{pi}{2}right) ). I know that ( cosleft(frac{pi}{2}right) = 0 ) and ( sinleft(frac{pi}{2}right) = 1 ). Plugging those in, the equation simplifies to ( 0 - sinalpha times 1 = -sinalpha ). So, ( -sinalpha = frac{3}{5} ).This means ( sinalpha = -frac{3}{5} ). Okay, so the sine of alpha is negative three fifths. Now, I need to find ( sin2alpha ). I remember the double-angle formula for sine: ( sin2alpha = 2sinalpha cosalpha ). So, I need to find both ( sinalpha ) and ( cosalpha ).I already have ( sinalpha = -frac{3}{5} ). To find ( cosalpha ), I can use the Pythagorean identity: ( sin^2alpha + cos^2alpha = 1 ). Plugging in ( sinalpha ), we get ( left(-frac{3}{5}right)^2 + cos^2alpha = 1 ). That simplifies to ( frac{9}{25} + cos^2alpha = 1 ). Subtracting ( frac{9}{25} ) from both sides gives ( cos^2alpha = 1 - frac{9}{25} = frac{16}{25} ). Taking the square root, ( cosalpha = pmfrac{4}{5} ).Now, I need to determine the sign of ( cosalpha ). The problem states that ( -frac{pi}{2} < alpha < frac{pi}{2} ). In this interval, cosine is always positive because cosine is positive in the first and fourth quadrants, which correspond to angles between ( -frac{pi}{2} ) and ( frac{pi}{2} ). So, ( cosalpha = frac{4}{5} ).Now, plugging ( sinalpha ) and ( cosalpha ) into the double-angle formula: ( sin2alpha = 2 times left(-frac{3}{5}right) times frac{4}{5} ). Calculating that, we get ( 2 times -frac{12}{25} = -frac{24}{25} ).Let me double-check my steps to make sure I didn't make a mistake. Starting from ( cosleft(alpha + frac{pi}{2}right) = frac{3}{5} ), I used the cosine addition formula correctly, simplified it to ( -sinalpha = frac{3}{5} ), so ( sinalpha = -frac{3}{5} ). Then, using the Pythagorean identity, I found ( cosalpha = frac{4}{5} ) because alpha is in the range where cosine is positive. Finally, applying the double-angle formula, I calculated ( sin2alpha = -frac{24}{25} ). That seems correct.So, looking back at the options, D is ( -frac{24}{25} ), which matches my result. I think that's the right answer.