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

Revision as of 10:42, 11 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

Solution 3 (Law of Cosines)

Let $O$ be the center of the circle. Notice how $OC = OD = r$ , where $r$ is the radius of the circle. By applying the law of cosines on triangle $OCE$ , $r^2=CE^2+OE^2-2(CE)(OE)\cos{45}=CE^2+OE^2-(CE)(OE)\sqrt{2}$ . Similarly, by applying the law of cosines on triangle $ODE$ , $r^2=DE^2+OE^2-2(DE)(OE)\cos{135}=DE^2+OE^2+(DE)(OE)\sqrt{2}$ . By subtracting these two equations, we get $CE^2-DE^2-(CE)(OE)\sqrt{2}-(DE)(OE)\sqrt{2}=0$ . We can rearrange it to get $CE^2-DE^2=(CE)(OE)\sqrt{2}+(DE)(OE)\sqrt{2}=(CE+DE)(OE\sqrt{2})$ . Because both $CE$ and $DE$ are both positive, we can safely divide both sides by $(CE+DE)$ to obtain $CE-DE=OE\sqrt{2}$ . Because $OE = OB - BE = 5\sqrt{2} - 2\sqrt{5}$ , $(CE-DE)^2 = CE^2+DE^2 - 2(CE)(DE) = (OE\sqrt{2})^2 =2(5\sqrt{2} - 2\sqrt{5})^2 = 140 - 40\sqrt{10}$ . Through power of a point, we can find out that $(CE)(DE)=20\sqrt{10} - 20$ , so $CE^2+DE^2 = (CE-DE)^2+ 2(CE)(DE)= (140 - 40\sqrt{10}) + 2(20\sqrt{10} - 20) = 100$

~Math_Wiz_3.14

Video Solution

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

@@ Line 13: / Line 13: @@
 ~CT17
+==Solution 3 (Law of Cosines)==
+Let <math>O</math> be the center of the circle. Notice how <math>OC = OD = r</math>, where <math>r</math> is the radius of the circle. By applying the law of cosines on triangle <math>OCE</math>, <math>r^2=CE^2+OE^2-2(CE)(OE)\cos{45}=CE^2+OE^2-(CE)(OE)\sqrt{2}</math>. Similarly, by applying the law of cosines on triangle <math>ODE</math>, <math>r^2=DE^2+OE^2-2(DE)(OE)\cos{135}=DE^2+OE^2+(DE)(OE)\sqrt{2}</math>. By subtracting these two equations, we get <math>CE^2-DE^2-(CE)(OE)\sqrt{2}-(DE)(OE)\sqrt{2}=0</math>. We can rearrange it to get <math>CE^2-DE^2=(CE)(OE)\sqrt{2}+(DE)(OE)\sqrt{2}=(CE+DE)(OE\sqrt{2})</math>. Because both <math>CE</math> and <math>DE</math> are both positive, we can safely divide both sides by <math>(CE+DE)</math> to obtain <math>CE-DE=OE\sqrt{2}</math>. Because <math>OE = OB - BE = 5\sqrt{2} - 2\sqrt{5}</math>, <math>(CE-DE)^2 = CE^2+DE^2 - 2(CE)(DE) = (OE\sqrt{2})^2 =2(5\sqrt{2} - 2\sqrt{5})^2 = 140 - 40\sqrt{10}</math>. Through power of a point, we can find out that <math>(CE)(DE)=20\sqrt{10} - 20</math>, so <math>CE^2+DE^2 = (CE-DE)^2+ 2(CE)(DE)= (140 - 40\sqrt{10}) + 2(20\sqrt{10} - 20) = 100</math>
+~Math_Wiz_3.14
 ==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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