💡Okay, so I have this problem about two curves in the rectangular coordinate system, and I need to convert them into polar coordinates. Then, I have to find the range of the product of distances from the origin along a ray that intersects both curves. Let me try to break this down step by step.First, let's understand the given curves. Curve ( C_1 ) is given by the equation ( (x - 2)^2 + y^2 = 4 ). That looks like the equation of a circle. The standard form of a circle is ( (x - h)^2 + (y - k)^2 = r^2 ), where ( (h, k) ) is the center and ( r ) is the radius. So, for ( C_1 ), the center is at ( (2, 0) ) and the radius is 2. That makes sense.Curve ( C_2 ) is given parametrically with ( x = 2cosphi ) and ( y = sinphi ). Hmm, parametric equations. I remember that parametric equations can often be converted to Cartesian form by eliminating the parameter. Let me try that. From ( x = 2cosphi ), I can solve for ( cosphi ) as ( cosphi = x/2 ). Similarly, from ( y = sinphi ), ( sinphi = y ). Now, I know that ( cos^2phi + sin^2phi = 1 ), so substituting the expressions from above, we get ( (x/2)^2 + y^2 = 1 ). Simplifying, that's ( x^2/4 + y^2 = 1 ). So, ( C_2 ) is an ellipse centered at the origin with semi-major axis 2 along the x-axis and semi-minor axis 1 along the y-axis.Alright, now part (I) asks for the polar equations of ( C_1 ) and ( C_2 ). I need to convert these Cartesian equations into polar coordinates. Remember, in polar coordinates, ( x = rhocostheta ) and ( y = rhosintheta ). Let me handle each curve one by one.Starting with ( C_1 ): ( (x - 2)^2 + y^2 = 4 ). Let's substitute ( x ) and ( y ) with polar coordinates.So, ( ( rhocostheta - 2 )^2 + ( rhosintheta )^2 = 4 ). Let me expand this:( (rhocostheta - 2)^2 = rho^2cos^2theta - 4rhocostheta + 4 )And ( (rhosintheta)^2 = rho^2sin^2theta )Adding them together:( rho^2cos^2theta - 4rhocostheta + 4 + rho^2sin^2theta = 4 )Combine like terms:( rho^2(cos^2theta + sin^2theta) - 4rhocostheta + 4 = 4 )Since ( cos^2theta + sin^2theta = 1 ), this simplifies to:( rho^2 - 4rhocostheta + 4 = 4 )Subtract 4 from both sides:( rho^2 - 4rhocostheta = 0 )Factor out ( rho ):( rho(rho - 4costheta) = 0 )So, either ( rho = 0 ) or ( rho = 4costheta ). Since ( rho = 0 ) is just the origin, which is already included in the other solution, the polar equation for ( C_1 ) is ( rho = 4costheta ). That seems straightforward.Now, moving on to ( C_2 ): ( x^2/4 + y^2 = 1 ). Again, substituting ( x = rhocostheta ) and ( y = rhosintheta ):( (rhocostheta)^2 / 4 + (rhosintheta)^2 = 1 )Simplify each term:( rho^2cos^2theta / 4 + rho^2sin^2theta = 1 )Factor out ( rho^2 ):( rho^2 ( cos^2theta / 4 + sin^2theta ) = 1 )Let me combine the terms inside the parentheses. To do that, I can write ( cos^2theta / 4 ) as ( (1/4)cos^2theta ) and ( sin^2theta ) as ( (4/4)sin^2theta ). So, adding them together:( (1/4)cos^2theta + (4/4)sin^2theta = (1/4)(cos^2theta + 4sin^2theta) )So, the equation becomes:( rho^2 cdot (1/4)(cos^2theta + 4sin^2theta) = 1 )Multiply both sides by 4:( rho^2 (cos^2theta + 4sin^2theta) = 4 )Therefore, solving for ( rho^2 ):( rho^2 = frac{4}{cos^2theta + 4sin^2theta} )Hmm, that looks a bit complicated. Maybe I can factor out something or simplify it further. Let me see:( cos^2theta + 4sin^2theta = cos^2theta + sin^2theta + 3sin^2theta = 1 + 3sin^2theta )Ah, that's a good simplification. So, substituting back:( rho^2 = frac{4}{1 + 3sin^2theta} )Therefore, ( rho = frac{2}{sqrt{1 + 3sin^2theta}} ). Since ( rho ) is a distance, it's non-negative, so we don't need to worry about the negative root.So, summarizing part (I):- Polar equation of ( C_1 ): ( rho = 4costheta )- Polar equation of ( C_2 ): ( rho = frac{2}{sqrt{1 + 3sin^2theta}} )Alright, that seems done. Now, moving on to part (II). We have a ray ( l ) with polar equation ( theta = alpha ), where ( 0 leq alpha leq frac{pi}{4} ) and ( rho > 0 ). This ray intersects ( C_1 ) at point ( M ) and ( C_2 ) at point ( N ), both different from the pole (which is the origin). We need to find the range of ( |OM| cdot |ON| ).So, essentially, for each angle ( alpha ) between 0 and ( pi/4 ), we draw a ray from the origin at that angle, find where it intersects ( C_1 ) and ( C_2 ), compute the distances from the origin to those points, multiply them, and then find the minimum and maximum values of this product as ( alpha ) varies from 0 to ( pi/4 ).Let me think about how to approach this.First, since both curves are given in polar form, and the ray is given by ( theta = alpha ), we can substitute ( theta = alpha ) into the polar equations of ( C_1 ) and ( C_2 ) to find ( rho ) for each curve, which will give us the distances ( |OM| ) and ( |ON| ).So, for ( C_1 ): ( rho = 4costheta ). Substituting ( theta = alpha ), we get ( |OM| = 4cosalpha ).For ( C_2 ): ( rho = frac{2}{sqrt{1 + 3sin^2theta}} ). Substituting ( theta = alpha ), we get ( |ON| = frac{2}{sqrt{1 + 3sin^2alpha}} ).Therefore, the product ( |OM| cdot |ON| ) is:( 4cosalpha cdot frac{2}{sqrt{1 + 3sin^2alpha}} = frac{8cosalpha}{sqrt{1 + 3sin^2alpha}} )So, now we have the expression for the product as a function of ( alpha ):( P(alpha) = frac{8cosalpha}{sqrt{1 + 3sin^2alpha}} )We need to find the range of ( P(alpha) ) as ( alpha ) varies from 0 to ( pi/4 ).To find the range, we can analyze the function ( P(alpha) ) over the interval ( [0, pi/4] ). Since ( P(alpha) ) is continuous on this interval, its extrema will occur either at critical points or at the endpoints.So, let's compute ( P(0) ) and ( P(pi/4) ), and then check if there are any critical points in between.First, ( P(0) ):( cos(0) = 1 ), ( sin(0) = 0 ). So,( P(0) = frac{8 cdot 1}{sqrt{1 + 0}} = frac{8}{1} = 8 )Next, ( P(pi/4) ):( cos(pi/4) = frac{sqrt{2}}{2} ), ( sin(pi/4) = frac{sqrt{2}}{2} ). So,( P(pi/4) = frac{8 cdot frac{sqrt{2}}{2}}{sqrt{1 + 3 cdot left( frac{sqrt{2}}{2} right)^2}} )Simplify numerator and denominator:Numerator: ( 8 cdot frac{sqrt{2}}{2} = 4sqrt{2} )Denominator: ( sqrt{1 + 3 cdot frac{2}{4}} = sqrt{1 + frac{6}{4}} = sqrt{1 + frac{3}{2}} = sqrt{frac{5}{2}} = frac{sqrt{10}}{2} )So, ( P(pi/4) = frac{4sqrt{2}}{sqrt{10}/2} = frac{4sqrt{2} cdot 2}{sqrt{10}} = frac{8sqrt{2}}{sqrt{10}} )Simplify ( sqrt{2}/sqrt{10} = sqrt{2/10} = sqrt{1/5} = 1/sqrt{5} ). So,( P(pi/4) = 8 cdot frac{1}{sqrt{5}} = frac{8}{sqrt{5}} = frac{8sqrt{5}}{5} )So, at the endpoints, ( P(0) = 8 ) and ( P(pi/4) = frac{8sqrt{5}}{5} approx 3.577 ). So, the function decreases from 8 to approximately 3.577 as ( alpha ) increases from 0 to ( pi/4 ).But we need to make sure if this function is strictly decreasing or if it has any critical points in between. So, let's find the derivative of ( P(alpha) ) with respect to ( alpha ) and see if it ever becomes zero or undefined in the interval ( (0, pi/4) ).Let me denote ( P(alpha) = frac{8cosalpha}{sqrt{1 + 3sin^2alpha}} ). Let's compute ( P'(alpha) ).First, write ( P(alpha) = 8cosalpha cdot (1 + 3sin^2alpha)^{-1/2} ). Let me use the product rule and chain rule for differentiation.Let me denote ( u = 8cosalpha ) and ( v = (1 + 3sin^2alpha)^{-1/2} ). Then, ( P = u cdot v ), so ( P' = u'v + uv' ).Compute ( u' ):( u = 8cosalpha Rightarrow u' = -8sinalpha )Compute ( v' ):( v = (1 + 3sin^2alpha)^{-1/2} )Let me set ( w = 1 + 3sin^2alpha ), so ( v = w^{-1/2} ). Then, ( dv/dw = (-1/2)w^{-3/2} ), and ( dw/dalpha = 6sinalphacosalpha ). Therefore, by the chain rule:( v' = dv/dalpha = dv/dw cdot dw/dalpha = (-1/2)w^{-3/2} cdot 6sinalphacosalpha = (-3sinalphacosalpha)w^{-3/2} )Substituting back ( w = 1 + 3sin^2alpha ):( v' = frac{ -3sinalphacosalpha }{ (1 + 3sin^2alpha)^{3/2} } )Now, putting it all together:( P' = u'v + uv' = (-8sinalpha)(1 + 3sin^2alpha)^{-1/2} + (8cosalpha)left( frac{ -3sinalphacosalpha }{ (1 + 3sin^2alpha)^{3/2} } right) )Let me factor out common terms:First term: ( -8sinalpha (1 + 3sin^2alpha)^{-1/2} )Second term: ( -24cos^2alpha sinalpha (1 + 3sin^2alpha)^{-3/2} )So, combining these:( P' = -8sinalpha (1 + 3sin^2alpha)^{-1/2} - 24cos^2alpha sinalpha (1 + 3sin^2alpha)^{-3/2} )To combine these terms, let me factor out ( -8sinalpha (1 + 3sin^2alpha)^{-3/2} ):Wait, let me see:First term: ( -8sinalpha (1 + 3sin^2alpha)^{-1/2} = -8sinalpha (1 + 3sin^2alpha)^{-1/2} cdot frac{(1 + 3sin^2alpha)}{(1 + 3sin^2alpha)} = -8sinalpha (1 + 3sin^2alpha)^{1/2} / (1 + 3sin^2alpha) )Wait, maybe that's complicating things. Alternatively, let me factor out ( -8sinalpha (1 + 3sin^2alpha)^{-3/2} ):So,( P' = -8sinalpha (1 + 3sin^2alpha)^{-3/2} [ (1 + 3sin^2alpha) + 3cos^2alpha ] )Let me verify:First term: ( -8sinalpha (1 + 3sin^2alpha)^{-1/2} = -8sinalpha (1 + 3sin^2alpha)^{-1/2} cdot (1 + 3sin^2alpha)^{-1} cdot (1 + 3sin^2alpha) )Wait, perhaps a better approach is to factor out ( -8sinalpha (1 + 3sin^2alpha)^{-3/2} ) from both terms:So,First term: ( -8sinalpha (1 + 3sin^2alpha)^{-1/2} = -8sinalpha (1 + 3sin^2alpha)^{-1/2} cdot (1 + 3sin^2alpha)^{-1} cdot (1 + 3sin^2alpha) )Wait, perhaps not. Let me instead factor out ( -8sinalpha (1 + 3sin^2alpha)^{-3/2} ):So,( P' = -8sinalpha (1 + 3sin^2alpha)^{-3/2} [ (1 + 3sin^2alpha) + 3cos^2alpha ] )Let me compute the expression inside the brackets:( (1 + 3sin^2alpha) + 3cos^2alpha = 1 + 3sin^2alpha + 3cos^2alpha = 1 + 3(sin^2alpha + cos^2alpha) = 1 + 3(1) = 4 )Wow, that's nice. So, the expression simplifies to:( P' = -8sinalpha (1 + 3sin^2alpha)^{-3/2} cdot 4 = -32sinalpha (1 + 3sin^2alpha)^{-3/2} )So, ( P' = -32sinalpha / (1 + 3sin^2alpha)^{3/2} )Now, since ( 0 leq alpha leq pi/4 ), ( sinalpha ) is non-negative (as sine is non-negative in the first quadrant). Also, ( (1 + 3sin^2alpha)^{3/2} ) is always positive. Therefore, ( P' ) is negative throughout the interval ( (0, pi/4) ).This means that ( P(alpha) ) is strictly decreasing on ( [0, pi/4] ). Therefore, the maximum value occurs at ( alpha = 0 ) and the minimum value occurs at ( alpha = pi/4 ).So, the range of ( |OM| cdot |ON| ) is from ( frac{8sqrt{5}}{5} ) to 8.Wait, let me just verify that ( frac{8sqrt{5}}{5} ) is indeed less than 8. Since ( sqrt{5} approx 2.236 ), so ( 8 times 2.236 / 5 approx 8 times 0.4472 approx 3.577 ), which is less than 8. So, that makes sense.Therefore, the range is ( left[ frac{8sqrt{5}}{5}, 8 right] ).Just to recap:- Converted both curves to polar coordinates.- Found expressions for ( |OM| ) and ( |ON| ) in terms of ( alpha ).- Computed the product ( |OM| cdot |ON| ) as a function of ( alpha ).- Analyzed the function by computing its derivative and found that it's strictly decreasing on the given interval.- Evaluated the function at the endpoints to find the maximum and minimum values.I think that covers everything. I don't see any mistakes in my reasoning, but let me just double-check the derivative calculation because that was a bit involved.Starting with ( P(alpha) = 8cosalpha / sqrt{1 + 3sin^2alpha} ). The derivative was calculated as ( P' = -32sinalpha / (1 + 3sin^2alpha)^{3/2} ). Let me recompute it quickly.Using the quotient rule: ( d/dalpha [f/g] = (f'g - fg') / g^2 ).Here, ( f = 8cosalpha ), so ( f' = -8sinalpha ).( g = sqrt{1 + 3sin^2alpha} ), so ( g' = (1/2)(1 + 3sin^2alpha)^{-1/2} cdot 6sinalphacosalpha = 3sinalphacosalpha / sqrt{1 + 3sin^2alpha} ).Therefore,( P' = [ (-8sinalpha) cdot sqrt{1 + 3sin^2alpha} - 8cosalpha cdot (3sinalphacosalpha / sqrt{1 + 3sin^2alpha}) ] / (1 + 3sin^2alpha) )Simplify numerator:First term: ( -8sinalpha sqrt{1 + 3sin^2alpha} )Second term: ( -24cos^2alpha sinalpha / sqrt{1 + 3sin^2alpha} )Factor out ( -8sinalpha / sqrt{1 + 3sin^2alpha} ):Numerator becomes:( -8sinalpha / sqrt{1 + 3sin^2alpha} [ (1 + 3sin^2alpha) + 3cos^2alpha ] )Which simplifies to:( -8sinalpha / sqrt{1 + 3sin^2alpha} cdot 4 ) as before.So, numerator is ( -32sinalpha / sqrt{1 + 3sin^2alpha} ), and denominator is ( 1 + 3sin^2alpha ). So, overall,( P' = (-32sinalpha) / (1 + 3sin^2alpha)^{3/2} ), which matches what I had earlier.So, the derivative is indeed negative throughout ( (0, pi/4) ), confirming that ( P(alpha) ) is strictly decreasing.Therefore, the range of ( |OM| cdot |ON| ) is from ( frac{8sqrt{5}}{5} ) to 8.**Final Answer**The range of ( |OM| cdot |ON| ) is boxed{left[ dfrac{8sqrt{5}}{5}, 8 right]}.

💡Okay, so I have this problem where I need to express ( x^6 + y^6 ) in terms of ( a ) and ( b ), given that ( x + y = a ) and ( xy = b ). Hmm, I remember that when dealing with expressions like this, it's often useful to use symmetric sums or identities that relate powers of variables to their sums and products. Let me think about how to approach this step by step.First, I know that ( x + y = a ) and ( xy = b ). I also recall that ( x^2 + y^2 ) can be expressed in terms of ( a ) and ( b ). Specifically, ( x^2 + y^2 = (x + y)^2 - 2xy ), which would be ( a^2 - 2b ). That seems straightforward.Next, maybe I can find ( x^3 + y^3 ) using the identity ( x^3 + y^3 = (x + y)(x^2 - xy + y^2) ). Substituting the known values, that would be ( a times (x^2 + y^2 - xy) ). Since I already have ( x^2 + y^2 = a^2 - 2b ) and ( xy = b ), this becomes ( a times (a^2 - 2b - b) = a(a^2 - 3b) ), which simplifies to ( a^3 - 3ab ). Okay, so ( x^3 + y^3 = a^3 - 3ab ).Now, I need to get to ( x^6 + y^6 ). I remember that ( x^6 + y^6 ) can be expressed as ( (x^3)^2 + (y^3)^2 ). There's an identity for the square of sums: ( (x^3 + y^3)^2 = x^6 + 2x^3y^3 + y^6 ). So, if I can find ( (x^3 + y^3)^2 ) and subtract ( 2x^3y^3 ), I should get ( x^6 + y^6 ).Let me compute ( (x^3 + y^3)^2 ) first. From earlier, ( x^3 + y^3 = a^3 - 3ab ), so squaring that gives ( (a^3 - 3ab)^2 ). Expanding this, I get ( a^6 - 6a^4b + 9a^2b^2 ). Next, I need to find ( 2x^3y^3 ). Since ( xy = b ), ( x^3y^3 = (xy)^3 = b^3 ). Therefore, ( 2x^3y^3 = 2b^3 ).Putting it all together, ( x^6 + y^6 = (a^3 - 3ab)^2 - 2b^3 ). Substituting the expanded form, that's ( a^6 - 6a^4b + 9a^2b^2 - 2b^3 ).Wait, let me double-check that. I squared ( a^3 - 3ab ) correctly? Let's see: ( (a^3)^2 = a^6 ), ( 2 times a^3 times (-3ab) = -6a^4b ), and ( (-3ab)^2 = 9a^2b^2 ). Yes, that seems right. Then subtracting ( 2b^3 ) gives the final expression.So, after working through the identities step by step, I arrive at the polynomial ( a^6 - 6a^4b + 9a^2b^2 - 2b^3 ) for ( x^6 + y^6 ).

💡Okay, so I have this problem here about the ratio of the areas of two squares being 250 over 98. I need to find the ratio of their side lengths, rationalize the denominator, and express it in the form a times the square root of b over c, where a, b, and c are integers. Then, I have to find the sum of a, b, and c. Hmm, let me think about how to approach this.First, I remember that the area of a square is equal to the side length squared. So, if I have two squares with areas A1 and A2, their side lengths would be sqrt(A1) and sqrt(A2), right? Therefore, the ratio of their side lengths would be sqrt(A1/A2). So, in this case, the ratio of the areas is 250/98, so the ratio of the side lengths should be sqrt(250/98).Wait, before I jump into taking the square root, maybe I should simplify the fraction 250 over 98 first. Let me see if I can reduce that. The numerator is 250, and the denominator is 98. Let's see if both numbers have a common factor. 250 divided by 2 is 125, and 98 divided by 2 is 49. So, 250 over 98 simplifies to 125 over 49. That's better because 125 and 49 don't have any common factors besides 1, so that's the simplest form.Okay, so now the ratio of the areas is 125 over 49. Therefore, the ratio of the side lengths is sqrt(125/49). I can separate the square root into the numerator and the denominator, so that becomes sqrt(125) over sqrt(49). I know that sqrt(49) is 7, so that's straightforward. Now, sqrt(125) is a bit trickier. Let me break that down.125 is 25 times 5, right? So, sqrt(125) is sqrt(25*5). I can separate that into sqrt(25) times sqrt(5), which is 5 times sqrt(5). So, putting that together, sqrt(125) is 5 sqrt(5). Therefore, the ratio of the side lengths is 5 sqrt(5) over 7.Wait, but the problem says to rationalize the denominator. Hmm, in this case, the denominator is already a rational number, 7, so I don't think I need to do anything else. But just to make sure, rationalizing the denominator usually means getting rid of any square roots in the denominator. In this case, the denominator is 7, which is already rational, so I think we're good.So, the ratio of the side lengths is 5 sqrt(5) over 7. Now, the problem wants it expressed in the form a sqrt(b) over c, where a, b, and c are integers. Let me check: a is 5, b is 5, and c is 7. So, that fits the form perfectly.Now, I need to find the sum of a, b, and c. So, that would be 5 + 5 + 7. Let me add that up: 5 plus 5 is 10, and 10 plus 7 is 17. So, the sum is 17.Wait a second, let me double-check my steps to make sure I didn't make a mistake. Starting from the beginning: the area ratio is 250/98, which simplifies to 125/49. Taking the square root gives sqrt(125)/sqrt(49), which is 5 sqrt(5)/7. Rationalizing the denominator isn't necessary here because there's no square root in the denominator. So, the expression is already in the desired form, and the sum is 5 + 5 + 7, which is indeed 17.I think that's correct. But just to be thorough, let me verify each step again. Simplifying 250/98: 250 divided by 2 is 125, and 98 divided by 2 is 49. Correct. Square root of 125 is 5 sqrt(5), and square root of 49 is 7. So, 5 sqrt(5)/7 is the ratio of the side lengths. Since the denominator is rational, no further rationalization is needed. Therefore, a is 5, b is 5, c is 7, and their sum is 17.Yeah, I think that's solid. I don't see any mistakes in my reasoning. So, the final answer should be 17.

💡First, I need to determine the total number of possible outcomes when tossing two fair coins. Each coin has two possible outcomes: Heads (H) or Tails (T). Therefore, for two coins, the possible outcomes are (H, H), (T, T), (H, T), and (T, H), making a total of 4 outcomes.Next, I identify the outcomes that result in exactly one head and one tail. These outcomes are (H, T) and (T, H).There are 2 favorable outcomes out of the total 4 possible outcomes. To find the probability, I divide the number of favorable outcomes by the total number of possible outcomes.So, the probability is 2 divided by 4, which simplifies to 1/2.Therefore, the probability of getting exactly one head and one tail when tossing two fair coins is 1/2.

💡First, I recognize that the length of an arc in a circle is determined by the formula ( L = frac{theta}{360} times 2pi R ), where ( theta ) is the central angle in degrees and ( R ) is the radius of the circle.Given that a ( 60^circ ) arc on circle ( C ) has the same length as a ( 40^circ ) arc on circle ( D ), I can set up the equation:[frac{60}{360} times 2pi R_C = frac{40}{360} times 2pi R_D]Simplifying this equation, I find that the ratio of the radii ( R_C ) to ( R_D ) is ( frac{2}{3} ).Since the area of a circle is proportional to the square of its radius, the ratio of the areas of circle ( C ) to circle ( D ) is:[left( frac{R_C}{R_D} right)^2 = left( frac{2}{3} right)^2 = frac{4}{9}]

💡Alright, so I'm trying to solve this geometry problem about a rectangle ABCD. Let me try to visualize it first. Since it's a rectangle, opposite sides are equal and all angles are right angles. The problem mentions a trapezoid ABCE and a triangle ADE. I think point E must be somewhere on side DC because otherwise, it wouldn't form a trapezoid ABCE. The problem gives me a few pieces of information:1. AD - AB = 9 cm. Since ABCD is a rectangle, AD and AB are the lengths of the sides. Let me denote AB as the length and AD as the width for clarity. So, width - length = 9 cm. Hmm, that seems a bit counterintuitive because usually, the length is longer than the width, but maybe not in this case. I'll keep that in mind.2. The area of trapezoid ABCE is 5 times the area of triangle ADE. Okay, so I need to figure out the areas of these two shapes. I know the formula for the area of a trapezoid is (1/2)*(sum of the two parallel sides)*height, and the area of a triangle is (1/2)*base*height.3. The perimeter of triangle ADE is 68 cm shorter than the perimeter of trapezoid ABCE. That's a significant difference, so I need to calculate both perimeters and set up an equation based on that.Let me start by assigning variables to the sides. Let me let AB = x cm. Then, since AD - AB = 9 cm, AD = x + 9 cm. So, the rectangle has sides of length x and x + 9.Now, I need to figure out where point E is located. Since ABCE is a trapezoid, sides AB and CE must be the two parallel sides. That means E is somewhere along DC. Let me denote DE as y cm. Then, EC would be DC - DE. Since DC is equal to AB, which is x cm, EC = x - y cm.Now, let's find the areas. The area of trapezoid ABCE is (1/2)*(AB + CE)*height. The height here would be AD, which is x + 9 cm. So, area of ABCE = (1/2)*(x + (x - y))*(x + 9) = (1/2)*(2x - y)*(x + 9).The area of triangle ADE is (1/2)*base*height. The base is DE = y cm, and the height is AD = x + 9 cm. So, area of ADE = (1/2)*y*(x + 9).According to the problem, area of ABCE = 5 * area of ADE. So:(1/2)*(2x - y)*(x + 9) = 5*(1/2)*y*(x + 9)Simplify both sides:(2x - y)*(x + 9) = 5y*(x + 9)Since (x + 9) is common on both sides, I can divide both sides by (x + 9), assuming x + 9 ≠ 0, which it isn't because lengths are positive.So, 2x - y = 5yCombine like terms:2x = 6y => y = (2x)/6 = x/3So, DE = y = x/3 cm, and EC = x - y = x - x/3 = (2x)/3 cm.Alright, that's useful information. Now, moving on to the perimeters.Perimeter of triangle ADE: It has sides AD, DE, and AE.AD = x + 9 cmDE = x/3 cmAE: Hmm, AE is the diagonal from A to E. Since E is on DC, which is of length x, and DE = x/3, then EC = 2x/3. So, coordinates might help here. Let me assign coordinates to the rectangle to find AE.Let me place point A at (0, 0). Then, since AB is x cm, point B is at (x, 0). AD is x + 9 cm, so point D is at (0, x + 9). Therefore, point C is at (x, x + 9). Point E is on DC, which goes from D(0, x + 9) to C(x, x + 9). Since DE = x/3, E is located at (x/3, x + 9).So, point E is at (x/3, x + 9). Therefore, AE is the distance from A(0,0) to E(x/3, x + 9). Using the distance formula:AE = sqrt[(x/3 - 0)^2 + (x + 9 - 0)^2] = sqrt[(x^2)/9 + (x + 9)^2]Let me compute that:= sqrt[(x^2)/9 + x^2 + 18x + 81]= sqrt[(x^2)/9 + (9x^2)/9 + 18x + 81]= sqrt[(10x^2)/9 + 18x + 81]Hmm, that seems a bit complicated. Maybe I can factor it differently or see if it simplifies.Alternatively, maybe I can express AE in terms of x without expanding. Let me see.Perimeter of triangle ADE = AD + DE + AE = (x + 9) + (x/3) + sqrt[(x/3)^2 + (x + 9)^2]Similarly, the perimeter of trapezoid ABCE: It has sides AB, BC, CE, and EA.AB = xBC = AD = x + 9CE = (2x)/3EA = same as above, sqrt[(x/3)^2 + (x + 9)^2]So, perimeter of ABCE = AB + BC + CE + EA = x + (x + 9) + (2x)/3 + sqrt[(x/3)^2 + (x + 9)^2]According to the problem, perimeter of ADE is 68 cm shorter than perimeter of ABCE. So:Perimeter of ABCE - Perimeter of ADE = 68Let me write that out:[x + (x + 9) + (2x)/3 + sqrt[(x/3)^2 + (x + 9)^2]] - [(x + 9) + (x/3) + sqrt[(x/3)^2 + (x + 9)^2]] = 68Simplify the left side:First, distribute the subtraction:x + (x + 9) + (2x)/3 + sqrt[(x/3)^2 + (x + 9)^2] - x - 9 - (x/3) - sqrt[(x/3)^2 + (x + 9)^2] = 68Now, let's cancel terms:- The sqrt terms cancel out.- x cancels with -x.- (x + 9) cancels with -9.- (2x)/3 - (x)/3 = (x)/3So, we're left with:(x)/3 = 68Therefore, x = 68 * 3 = 204 cmWait, that seems quite large. Let me double-check my steps.Starting from the perimeters:Perimeter of ABCE: x + (x + 9) + (2x)/3 + AEPerimeter of ADE: (x + 9) + (x/3) + AESubtracting them:[x + (x + 9) + (2x)/3 + AE] - [(x + 9) + (x/3) + AE] = x + (x + 9) + (2x)/3 + AE - x - 9 - x/3 - AESimplify term by term:x - x = 0(x + 9) - (x + 9) = 0(2x)/3 - x/3 = x/3AE - AE = 0So, indeed, it's x/3 = 68 => x = 204 cmHmm, 204 cm seems very long for a rectangle, but mathematically, it's consistent. Let me check if this makes sense with the other information.We have AB = x = 204 cmAD = x + 9 = 213 cmSo, the rectangle is 204 cm by 213 cm. That's a huge rectangle, but let's see if the areas make sense.Area of rectangle ABCD = AB * AD = 204 * 213. Let me compute that:204 * 200 = 40,800204 * 13 = 2,652Total area = 40,800 + 2,652 = 43,452 cm²Now, area of triangle ADE: (1/2)*DE*AD = (1/2)*(x/3)*(x + 9) = (1/2)*(204/3)*(213) = (1/2)*(68)*(213)Compute 68 * 213:68 * 200 = 13,60068 * 13 = 884Total = 13,600 + 884 = 14,484Half of that is 7,242 cm²Area of trapezoid ABCE = 5 * area of ADE = 5 * 7,242 = 36,210 cm²Now, let's compute the area of the trapezoid ABCE using the formula:(1/2)*(AB + CE)*AD = (1/2)*(204 + (2*204)/3)*213Compute CE = (2x)/3 = (2*204)/3 = 136 cmSo, (1/2)*(204 + 136)*213 = (1/2)*(340)*213 = 170 * 213Compute 170 * 200 = 34,000170 * 13 = 2,210Total = 34,000 + 2,210 = 36,210 cm²That matches the given condition, so the area calculations are consistent.Now, checking the perimeters:Perimeter of ADE: AD + DE + AE = 213 + 68 + AECompute AE: sqrt[(x/3)^2 + (x + 9)^2] = sqrt[(204/3)^2 + (213)^2] = sqrt[68² + 213²]Compute 68² = 4,624213² = 45,369So, AE = sqrt[4,624 + 45,369] = sqrt[49,993] ≈ 223.6 cmSo, perimeter of ADE ≈ 213 + 68 + 223.6 ≈ 504.6 cmPerimeter of ABCE: AB + BC + CE + EA = 204 + 213 + 136 + 223.6 ≈ 204 + 213 = 417; 136 + 223.6 = 359.6; total ≈ 417 + 359.6 ≈ 776.6 cmDifference: 776.6 - 504.6 ≈ 272 cmWait, that's not 68 cm. There's a discrepancy here. Did I make a mistake?Wait, earlier when I subtracted the perimeters, I got x/3 = 68, so x = 204. But when I compute the actual perimeters, the difference is about 272 cm, not 68 cm. That means I must have made a mistake in my earlier reasoning.Let me go back to the perimeter difference.Perimeter of ABCE - Perimeter of ADE = 68But when I calculated, it was 776.6 - 504.6 ≈ 272 cm, which is much larger than 68 cm. So, where did I go wrong?Wait, let's re-examine the perimeters.Perimeter of ABCE: AB + BC + CE + EAAB = xBC = AD = x + 9CE = (2x)/3EA = sqrt[(x/3)^2 + (x + 9)^2]Perimeter of ADE: AD + DE + AE = (x + 9) + (x/3) + sqrt[(x/3)^2 + (x + 9)^2]So, when subtracting:Perimeter ABCE - Perimeter ADE = [x + (x + 9) + (2x)/3 + sqrt(...)] - [(x + 9) + (x/3) + sqrt(...)] = x + (x + 9) + (2x)/3 - (x + 9) - (x)/3 = x + (2x)/3 - x/3 = x + (x)/3 = (4x)/3Wait, earlier I thought it was x/3, but actually, it's (4x)/3 = 68So, (4x)/3 = 68 => x = (68 * 3)/4 = 51 cmAh, that's where I made a mistake earlier. I incorrectly simplified the subtraction. It should be (4x)/3 = 68, not x/3 = 68.So, x = (68 * 3)/4 = 51 cmOkay, that makes more sense. So, AB = x = 51 cmAD = x + 9 = 60 cmSo, the rectangle is 51 cm by 60 cm.Now, let's recalculate the areas.Area of rectangle ABCD = 51 * 60 = 3,060 cm²Area of triangle ADE: (1/2)*DE*AD = (1/2)*(x/3)*(x + 9) = (1/2)*(51/3)*(60) = (1/2)*(17)*(60) = 8.5 * 60 = 510 cm²Area of trapezoid ABCE = 5 * area of ADE = 5 * 510 = 2,550 cm²Let me verify the area of trapezoid ABCE using the formula:(1/2)*(AB + CE)*AD = (1/2)*(51 + (2*51)/3)*60 = (1/2)*(51 + 34)*60 = (1/2)*(85)*60 = 42.5 * 60 = 2,550 cm²That matches, so the areas are consistent.Now, let's check the perimeters with x = 51 cm.Compute AE: sqrt[(x/3)^2 + (x + 9)^2] = sqrt[(51/3)^2 + (60)^2] = sqrt[17² + 60²] = sqrt[289 + 3,600] = sqrt[3,889] ≈ 62.37 cmPerimeter of ADE: AD + DE + AE = 60 + 17 + 62.37 ≈ 139.37 cmPerimeter of ABCE: AB + BC + CE + EA = 51 + 60 + 34 + 62.37 ≈ 51 + 60 = 111; 34 + 62.37 = 96.37; total ≈ 111 + 96.37 ≈ 207.37 cmDifference: 207.37 - 139.37 ≈ 68 cmYes, that matches the given condition. So, the correct value of x is 51 cm, making AD = 60 cm, and the area of the rectangle is 51 * 60 = 3,060 cm².I think that makes sense now. Initially, I made a mistake in the perimeter subtraction, but after correcting that, everything aligns properly.