Problem 268

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Problem 268: (2021 AMC 12b,17)

Let $ABCD$ be an isosceles trapezoid having parallel bases $\overline{AB}$ and $\overline{CD}$ with $AB>CD.$ Line segments from a point inside $ABCD$ to the vertices divide the trapezoid into four triangles whose areas are $2, 3, 4,$ and $5$ starting with the triangle with base $\overline{CD}$ and moving clockwise as shown in the diagram below. What is the ratio $\frac{AB}{CD}?$

$\textbf{(A)}\: 3\qquad$ $\textbf{(B)}\: 2+\sqrt{2}\qquad$ $\textbf{(C)}\: 1+\sqrt{6}\qquad$ $\textbf{(D)}\: 2\sqrt{3}\qquad$ $\textbf{(E)}\: 3\sqrt{2}$

Solution

Exercise 268: Solution

Let $ABCD$ be an isosceles trapezoid having parallel bases $\overline{AB}$ and $\overline{CD}$ with $AB>CD.$ Line segments from a point inside $ABCD$ to the vertices divide the trapezoid into four triangles whose areas are $1, 2, 3,$ and $4,$ starting with the triangle with base $\overline{CD}$ and moving clockwise. What is $\frac{AB}{CD}?$

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