Difference between revisions of "2020 AMC 12B Problems/Problem 12"

Revision as of 22:18, 10 February 2020

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 1

Let $O$ be the center of the circle, and $X$ be the midpoint of $\overline{CD}$ . Let $CX=a$ and $EX=b$ . This implies that $DE = a - b$ . Since $CE = CX + EX = a + b$ , 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 $a^2 + b^2 = CX^2 + OX^2 = (5\sqrt{2})^2=50$ . Thus, our answer is $2(50)=\boxed{\textbf{(E) } 100}$ .

~JHawk0224

Solution 2 (Power of a Point)

Let $O$ be the center of the circle, and $X$ be the midpoint of $CD$ . Draw triangle $OCD$ , and median $OX$ . Because $OC = OD$ , $OCD$ is isosceles, so $OX$ is also an altitude of $OCD$ . $OD = 5\sqrt2 - 2\sqrt5$ , and because angle $OEC$ is $45$ degrees and triangle $OXE$ is right, $OX = EX = \frac{5\sqrt2 - 2\sqrt5}{\sqrt2} = 5 - \sqrt{10}$ . Because triangle $OXC$ is right, $CX = \sqrt{(5\sqrt2)^2 - (5 - \sqrt{10})^2} = \sqrt{15 + 10\sqrt{10}}$ . Thus, $CD = 2\sqrt{15 + 10\sqrt{10}}$ . We are looking for $CE^2$ + $DE^2$ which is also $(CE + DE)^2 - 2 \cdot CE \cdot DE$ . Because $CE + DE = CD = 2\sqrt{15 + 10\sqrt{10}}, (CE + CD)^2 = 4(15 + 10\sqrt{10}) = 60 + 40\sqrt{10}$ . By power of a point, $CE \cdot DE = AE \cdot BE = 2\sqrt5\cdot(10\sqrt2 - 2\sqrt5) = 20\sqrt{10} - 20$ so $2 \cdot CE \cdot DE = 40\sqrt{10} - 40$ . Finally, $CE^2 + DE^2 = 60 + 40\sqrt{10} - (40\sqrt{10} - 40) = \boxed{(E) 100}$ .

~CT17

Video Solution

On The Spot STEM: https://www.youtube.com/watch?v=h-hhRa93lK4

@@ Line 13: / Line 13: @@
 ~CT17
-==Solution 3(bash)==
 ==Video Solution==
 On The Spot STEM: https://www.youtube.com/watch?v=h-hhRa93lK4

2020 AMC 12B (Problems • Answer Key • Resources)
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