2020 AMC 12B Problems/Problem 12

Problem

Let $\overline{AB}$ be a diameter in a circle of radius $5\sqrt2.$ Let $\overline{CD}$ be a chord in the circle that intersects $\overline{AB}$ at a point $E$ such that $BE=2\sqrt5$ and $\angle AEC = 45^{\circ}.$ What is $CE^2+DE^2?$

$\textbf{(A)}\ 96 \qquad\textbf{(B)}\ 98 \qquad\textbf{(C)}\ 44\sqrt5 \qquad\textbf{(D)}\ 70\sqrt2 \qquad\textbf{(E)}\ 100$

Solution

Let $O$ be the center of the circle, and $X$ be the midpoint of $\overline{CD}$ . Let $XC=a$ and $XE=b$ . This implies that $ED = a - b$ . Since $CE = XC + XE = a + b$ , we now want to find $(a+b)^2+(a-b)^2=2(a^2+b^2)$ . Since $\angle CXO$ is a right angle, by Pythagorean theorem $a^2 + b^2 = XC^2 + XO^2 = (5\sqrt{2})^2=50$ . Thus, our answer is $2(50)=\boxed{\textbf{(E) } 100}$ .

2020 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
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All AMC 12 Problems and Solutions

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2020 AMC 12B Problems/Problem 12

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