💡Alright, so I have this problem about square ABCD with side length 30. There's a point P inside the square such that AP is 12 and BP is 26. I need to find the area of the quadrilateral formed by the centroids of triangles ABP, BCP, CDP, and DAP. Hmm, okay, let me try to break this down step by step.First, I know that the centroid of a triangle is the intersection point of its three medians, and it divides each median into a ratio of 2:1, with the longer part being closer to the vertex. So, for each triangle mentioned, the centroid will be located at the average of the coordinates of the three vertices.Since ABCD is a square, I can assign coordinates to the vertices to make things easier. Let me place point A at (0, 30), B at (0, 0), C at (30, 0), and D at (30, 30). This way, the square is in the first quadrant with side length 30.Now, point P is inside the square. I know AP = 12 and BP = 26. I need to find the coordinates of P. Let me denote P as (x, y). Then, using the distance formula:AP = 12: The distance from A(0, 30) to P(x, y) is sqrt[(x - 0)^2 + (y - 30)^2] = 12.So, (x)^2 + (y - 30)^2 = 144. (1)Similarly, BP = 26: The distance from B(0, 0) to P(x, y) is sqrt[(x - 0)^2 + (y - 0)^2] = 26.So, (x)^2 + (y)^2 = 676. (2)Now, I can subtract equation (1) from equation (2) to eliminate x^2:(x)^2 + y^2 - [(x)^2 + (y - 30)^2] = 676 - 144Simplify:x^2 + y^2 - x^2 - (y^2 - 60y + 900) = 532So, y^2 - y^2 + 60y - 900 = 532Simplify:60y - 900 = 53260y = 1432y = 1432 / 60Simplify:Divide numerator and denominator by 4: 358 / 15 ≈ 23.8667So, y ≈ 23.8667.Now, plug y back into equation (2) to find x:x^2 + (358/15)^2 = 676Calculate (358/15)^2:358 squared is 128,164. Divided by 225: 128,164 / 225 ≈ 569.6622So, x^2 ≈ 676 - 569.6622 ≈ 106.3378Therefore, x ≈ sqrt(106.3378) ≈ 10.3125So, point P is approximately at (10.3125, 23.8667). Let me note that as (10.3125, 23.8667).Now, I need to find the centroids of the four triangles: ABP, BCP, CDP, and DAP.Let me recall that the centroid (G) of a triangle with vertices (x1, y1), (x2, y2), (x3, y3) is given by:G = [(x1 + x2 + x3)/3, (y1 + y2 + y3)/3]So, let's compute each centroid.1. Centroid of triangle ABP:A is (0, 30), B is (0, 0), P is (10.3125, 23.8667)Centroid C1:x = (0 + 0 + 10.3125)/3 ≈ 10.3125 / 3 ≈ 3.4375y = (30 + 0 + 23.8667)/3 ≈ 53.8667 / 3 ≈ 17.9556So, C1 ≈ (3.4375, 17.9556)2. Centroid of triangle BCP:B is (0, 0), C is (30, 0), P is (10.3125, 23.8667)Centroid C2:x = (0 + 30 + 10.3125)/3 ≈ 40.3125 / 3 ≈ 13.4375y = (0 + 0 + 23.8667)/3 ≈ 23.8667 / 3 ≈ 7.9556So, C2 ≈ (13.4375, 7.9556)3. Centroid of triangle CDP:C is (30, 0), D is (30, 30), P is (10.3125, 23.8667)Centroid C3:x = (30 + 30 + 10.3125)/3 ≈ 70.3125 / 3 ≈ 23.4375y = (0 + 30 + 23.8667)/3 ≈ 53.8667 / 3 ≈ 17.9556So, C3 ≈ (23.4375, 17.9556)4. Centroid of triangle DAP:D is (30, 30), A is (0, 30), P is (10.3125, 23.8667)Centroid C4:x = (30 + 0 + 10.3125)/3 ≈ 40.3125 / 3 ≈ 13.4375y = (30 + 30 + 23.8667)/3 ≈ 83.8667 / 3 ≈ 27.9556So, C4 ≈ (13.4375, 27.9556)Now, I have the four centroids:C1 ≈ (3.4375, 17.9556)C2 ≈ (13.4375, 7.9556)C3 ≈ (23.4375, 17.9556)C4 ≈ (13.4375, 27.9556)Hmm, interesting. Let me plot these points mentally. C1 is near the left side, C2 is near the bottom, C3 is near the right side, and C4 is near the top. Connecting these should form a quadrilateral.Wait, looking at the coordinates, I notice that C1 and C3 have the same y-coordinate, approximately 17.9556, and C2 and C4 have x-coordinate approximately 13.4375. So, the quadrilateral is symmetric in some way.Let me compute the exact coordinates without approximating to see if there's a pattern.Wait, maybe I can find exact values instead of approximate decimals. Let me go back to the equations.From earlier, we had:From equation (2): x^2 + y^2 = 676From equation (1): x^2 + (y - 30)^2 = 144Subtracting (1) from (2):y^2 - (y - 30)^2 = 676 - 144Expanding (y - 30)^2: y^2 - 60y + 900So, y^2 - (y^2 - 60y + 900) = 532Simplify: 60y - 900 = 53260y = 1432y = 1432 / 60Simplify numerator and denominator by 4: 358 / 15So, y = 358/15Similarly, x^2 = 676 - y^2Compute y^2: (358/15)^2 = (358)^2 / 225358 squared: Let's compute 350^2 + 2*350*8 + 8^2 = 122500 + 5600 + 64 = 128,164So, y^2 = 128,164 / 225Thus, x^2 = 676 - 128,164 / 225Convert 676 to over 225: 676 * 225 = let's compute 676*200 + 676*25 = 135,200 + 16,900 = 152,100So, x^2 = (152,100 - 128,164) / 225 = (23,936) / 225Thus, x = sqrt(23,936 / 225) = sqrt(23,936)/15Compute sqrt(23,936): Let's see, 150^2 = 22,500, 160^2=25,600. So, between 150 and 160.Compute 154^2 = 23,716155^2=24,025So, 154^2=23,716, which is less than 23,936.Difference: 23,936 - 23,716 = 220So, 154 + 220/(2*154 +1) ≈ 154 + 220/309 ≈ 154 + 0.712 ≈ 154.712But since we have sqrt(23,936)/15, it's approximately 154.712 /15 ≈ 10.314Which matches our earlier approximation.So, exact coordinates of P are (sqrt(23,936)/15, 358/15). Hmm, but maybe I can write sqrt(23,936) as 154.712, but perhaps it's better to keep it symbolic.Alternatively, maybe I can express x and y in fractions.Wait, 23,936 divided by 16 is 1,496. So, sqrt(23,936) = 4*sqrt(1,496). Hmm, not sure if that helps.Alternatively, maybe I can leave it as is for now.But perhaps instead of working with approximate decimals, I can work with exact fractions.Let me try that.So, point P is (sqrt(23,936)/15, 358/15). Let me denote sqrt(23,936) as S for simplicity.So, P = (S/15, 358/15)Now, let's compute the centroids exactly.1. Centroid C1 of triangle ABP:A = (0, 30), B = (0, 0), P = (S/15, 358/15)Centroid C1:x = (0 + 0 + S/15)/3 = S/(15*3) = S/45y = (30 + 0 + 358/15)/3Convert 30 to 450/15:y = (450/15 + 358/15)/3 = (808/15)/3 = 808/(15*3) = 808/45So, C1 = (S/45, 808/45)2. Centroid C2 of triangle BCP:B = (0, 0), C = (30, 0), P = (S/15, 358/15)Centroid C2:x = (0 + 30 + S/15)/3 = (30 + S/15)/3 = 10 + S/45y = (0 + 0 + 358/15)/3 = (358/15)/3 = 358/45So, C2 = (10 + S/45, 358/45)3. Centroid C3 of triangle CDP:C = (30, 0), D = (30, 30), P = (S/15, 358/15)Centroid C3:x = (30 + 30 + S/15)/3 = (60 + S/15)/3 = 20 + S/45y = (0 + 30 + 358/15)/3Convert 30 to 450/15:y = (450/15 + 358/15)/3 = (808/15)/3 = 808/45So, C3 = (20 + S/45, 808/45)4. Centroid C4 of triangle DAP:D = (30, 30), A = (0, 30), P = (S/15, 358/15)Centroid C4:x = (30 + 0 + S/15)/3 = (30 + S/15)/3 = 10 + S/45y = (30 + 30 + 358/15)/3Convert 30 to 450/15:y = (450/15 + 450/15 + 358/15)/3 = (1258/15)/3 = 1258/45So, C4 = (10 + S/45, 1258/45)Now, let's write down all four centroids:C1 = (S/45, 808/45)C2 = (10 + S/45, 358/45)C3 = (20 + S/45, 808/45)C4 = (10 + S/45, 1258/45)Hmm, interesting. Let's see if we can find a pattern or relationship between these points.Looking at the x-coordinates:C1: S/45C2: 10 + S/45C3: 20 + S/45C4: 10 + S/45Similarly, y-coordinates:C1: 808/45C2: 358/45C3: 808/45C4: 1258/45So, the x-coordinates of C1, C2, C3, C4 are S/45, 10 + S/45, 20 + S/45, 10 + S/45Similarly, y-coordinates are 808/45, 358/45, 808/45, 1258/45Wait, so C1 and C3 have the same y-coordinate, 808/45, and C2 and C4 have x-coordinate 10 + S/45.Also, C1 and C3 are symmetric around the vertical line x = 10 + S/45, since C1 is at S/45 and C3 is at 20 + S/45, which is 10 + S/45 + 10.Similarly, C2 and C4 are symmetric around the horizontal line y = 808/45, since C2 is at 358/45 and C4 is at 1258/45, which is 808/45 + 450/45 = 808/45 + 10.Wait, 1258 - 808 = 450, so 450/45 = 10.So, the quadrilateral formed by C1, C2, C3, C4 is symmetric both vertically and horizontally.Therefore, it's a parallelogram, and more specifically, a rectangle, because the sides are perpendicular.But let me verify that.Wait, actually, since the figure is symmetric about both the vertical and horizontal lines through the center, it's likely a rectangle.But let me compute the coordinates more precisely.Wait, let me note that S is sqrt(23,936). Let me compute that exactly.23,936 divided by 16 is 1,496.1,496 divided by 16 is 93.5. Hmm, not a perfect square.Wait, maybe 23,936 = 16 * 1,4961,496 divided by 4 is 374374 divided by 2 is 187187 is 11*17So, 23,936 = 16 * 4 * 2 * 11 * 17 = 16*4*2*11*17 = 16*4=64, 64*2=128, 128*11=1,408, 1,408*17=23,936So, sqrt(23,936) = sqrt(16 * 1,496) = 4*sqrt(1,496)Similarly, 1,496 = 4*374, so sqrt(1,496) = 2*sqrt(374)374 = 2*187, and 187 is 11*17, which are primes.So, sqrt(23,936) = 4*2*sqrt(374) = 8*sqrt(374)But 374 is 2*11*17, so it's not a perfect square.Therefore, S = 8*sqrt(374). Hmm, that's as simplified as it gets.But perhaps I can leave it as S for now.So, S = sqrt(23,936) = 8*sqrt(374)Therefore, S/45 = (8*sqrt(374))/45So, the coordinates are:C1 = (8*sqrt(374)/45, 808/45)C2 = (10 + 8*sqrt(374)/45, 358/45)C3 = (20 + 8*sqrt(374)/45, 808/45)C4 = (10 + 8*sqrt(374)/45, 1258/45)Now, to find the area of the quadrilateral formed by these four points.Since it's a convex quadrilateral, and given the symmetry, it's likely a parallelogram. To confirm, let's check if the midpoints of the diagonals coincide.The midpoint of C1C3 should be the same as the midpoint of C2C4.Compute midpoint of C1C3:x: [8*sqrt(374)/45 + 20 + 8*sqrt(374)/45]/2 = [20 + 16*sqrt(374)/45]/2 = 10 + 8*sqrt(374)/45y: [808/45 + 808/45]/2 = 808/45Midpoint of C2C4:x: [10 + 8*sqrt(374)/45 + 10 + 8*sqrt(374)/45]/2 = [20 + 16*sqrt(374)/45]/2 = 10 + 8*sqrt(374)/45y: [358/45 + 1258/45]/2 = [1616/45]/2 = 808/45So, both midpoints are the same: (10 + 8*sqrt(374)/45, 808/45). Therefore, the quadrilateral is a parallelogram.Now, to find the area of a parallelogram, we can use the base times height, or alternatively, the magnitude of the cross product of two adjacent sides.Alternatively, since it's a parallelogram, we can compute the vectors of two adjacent sides and take the magnitude of their cross product.Let me compute vectors C1C2 and C1C4.First, vector C1C2:C2 - C1 = [10 + 8*sqrt(374)/45 - 8*sqrt(374)/45, 358/45 - 808/45] = [10, -450/45] = [10, -10]Similarly, vector C1C4:C4 - C1 = [10 + 8*sqrt(374)/45 - 8*sqrt(374)/45, 1258/45 - 808/45] = [10, 450/45] = [10, 10]So, vectors C1C2 = (10, -10) and C1C4 = (10, 10)The area of the parallelogram is the magnitude of the cross product of these two vectors.The cross product in 2D is given by |a_x * b_y - a_y * b_x|So, |10*10 - (-10)*10| = |100 - (-100)| = |200| = 200Therefore, the area is 200.Wait, that's one of the answer choices, option C.But let me double-check my calculations because it seems straightforward, but I want to make sure I didn't make any mistakes.So, vectors from C1:C1C2: (10, -10)C1C4: (10, 10)Cross product: 10*10 - (-10)*10 = 100 + 100 = 200Yes, that seems correct.Alternatively, since the sides are (10, -10) and (10, 10), the area is |(10)(10) - (-10)(10)| = 200.Alternatively, since the sides are of length sqrt(10^2 + (-10)^2) = sqrt(200) and sqrt(10^2 + 10^2) = sqrt(200), and the angle between them is 90 degrees because the dot product is (10)(10) + (-10)(10) = 100 - 100 = 0, so they are perpendicular.Therefore, area = base * height = sqrt(200) * sqrt(200) = 200.Yes, that confirms it.So, the area of the quadrilateral formed by the centroids is 200.**Final Answer**boxed{C}

💡First, I need to understand what it means to find the opposite of a number. The opposite of a number is the value that, when added to the original number, results in zero. This means that the opposite of a positive number is negative, and the opposite of a negative number is positive.Given the number (-frac{1}{2}), I want to find its opposite. To do this, I will change the sign of the number. Changing the negative sign to a positive sign gives me (frac{1}{2}).To verify, I can add the original number and its opposite to see if the result is zero:[-frac{1}{2} + frac{1}{2} = 0]Since the sum is zero, (frac{1}{2}) is indeed the correct opposite of (-frac{1}{2}).Therefore, the correct answer is (boxed{text{A: } frac{1}{2}}).

💡I need to convert the damage amount from US dollars to Euros. The given damage is 45 million US dollars, and the exchange rate is 1 US dollar equals 0.9 Euros.First, I'll write down the conversion formula:[text{Damage in Euros} = text{Damage in US dollars} times text{Exchange rate}]Substituting the given values:[text{Damage in Euros} = 45,000,000 times 0.9]Now, I'll perform the multiplication:[45,000,000 times 0.9 = 40,500,000]So, the damage caused by the hurricane in Euros is 40,500,000.

💡Alright, so I've got this problem here: There's a set A that contains three elements: 0, -1, and x. It's also given that x squared is in A. I need to find the value of the real number x. Hmm, okay, let's break this down step by step.First, let's understand what the set A looks like. It has three elements: 0, -1, and x. So, A = {0, -1, x}. Now, it's given that x squared is also in A. That means x² must be one of the elements in the set A. So, x² can be either 0, -1, or x itself.Let me write that down:1. x² = 02. x² = -13. x² = xOkay, let's consider each case one by one.**Case 1: x² = 0**If x squared equals 0, then x must be 0. That seems straightforward. So, x = 0. But wait, if x is 0, then our set A becomes {0, -1, 0}. But sets cannot have duplicate elements, right? So, actually, A would just be {0, -1}, which only has two elements. But the problem states that A has three elements: 0, -1, and x. So, having x = 0 would reduce the number of distinct elements in the set, which contradicts the given information. Therefore, x cannot be 0. So, this case doesn't work.**Case 2: x² = -1**Hmm, x squared equals -1. But wait, x is a real number, right? Because the problem specifies that x is a real number. And in real numbers, the square of any real number is always non-negative. So, x² can't be negative. Therefore, x² = -1 has no real solutions. So, this case is impossible. We can disregard this case.**Case 3: x² = x**Okay, so x squared equals x. Let's solve this equation. If x² = x, then we can subtract x from both sides to get:x² - x = 0Factor out an x:x(x - 1) = 0So, this gives us two solutions: x = 0 or x = 1.Now, let's check these solutions in the context of our set A.- If x = 0, as we saw in Case 1, the set A would become {0, -1, 0}, which is just {0, -1}, having only two distinct elements. But the problem states that A has three elements, so x cannot be 0.- If x = 1, then our set A becomes {0, -1, 1}. That's three distinct elements, which fits the given condition. So, x = 1 seems to be a valid solution.Wait a second, but let me double-check. If x = 1, then x² = 1² = 1, which is indeed in set A. So, that works perfectly.Is there any other possibility? Well, we considered all three cases:1. x² = 0 leads to x = 0, which doesn't work because it reduces the number of elements.2. x² = -1 has no real solutions.3. x² = x leads to x = 0 or x = 1, and only x = 1 works.So, x must be 1.But just to make sure I haven't missed anything, let me think again. The set A has three elements: 0, -1, and x. x squared must be one of these. We've considered all possibilities, and only x = 1 satisfies all conditions without causing any contradictions.Therefore, I'm confident that the value of x is 1.**Final Answer**The value of ( x ) is boxed{1}.

💡First, identify the base cost of the cell phone plan, which is 25.Next, calculate the cost for text messages. Carl sent 150 text messages at a rate of 0.03 per text:150 texts * 0.03/text = 4.50.Then, determine the cost for extra minutes. Carl talked for 26 hours, which is 1 hour over the included 25 hours. Convert the extra hour to minutes:1 hour * 60 minutes/hour = 60 minutes.Calculate the additional cost for the extra minutes at 0.15 per minute:60 minutes * 0.15/minute = 9.00.Finally, add up all the costs:25 (base cost) + 4.50 (texts) + 9.00 (extra minutes) = 38.50.

💡First, I recognize that 30 fish were tagged and released back into the pond. Later, a second catch was made, and out of the fish caught in the second catch, 2 were tagged. The goal is to find out how many fish were caught in the second catch, given that the total approximate number of fish in the pond is 750.I understand that the percentage of tagged fish in the second catch should be approximately equal to the percentage of tagged fish in the entire pond. This means that the ratio of tagged fish in the second catch (2 out of the total caught in the second catch) should be similar to the ratio of tagged fish in the entire pond (30 out of 750).To find the number of fish caught in the second catch, I set up a proportion based on these ratios:[frac{2}{x} approx frac{30}{750}]Next, I solve for ( x ) by cross-multiplying:[2 cdot 750 approx 30 cdot x][1500 approx 30x]Finally, I divide both sides by 30 to find the value of ( x ):[x approx frac{1500}{30} = 50]Therefore, approximately 50 fish were caught in the second catch.