ABCE is an isosceles trapezoid and ACDE is a rectangle. AB = 10 and EC = 20. What is the length of AE?
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Answer / Prem Narayan Singh
Since ABCE is an isosceles trapezoid, BC = AB = 10. In a rectangle, the diagonal divides it into two right-angled triangles with equal base (AD) and height (DE). So, DE^2 + AD^2 = AC^2 (Pythagorean theorem). Since AC = EC - AB = 20 - 10 = 10, DE^2 + AD^2 = 10^2. Let's denote AD as x. Then, DE^2 = (10^2 - x^2)/4 (since diagonal of a right-angled triangle is square root of the sum of squares of two sides minus the square of the remaining side divided by 2). Since DE and AE are equal (they are parts of the same diagonal), AE^2 = x^2. Solving these equations, we find x = sqrt(100 - 82) = 4. So, AE = x = 4.
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