Difference between revisions of "2021 Fall AMC 10A Problems/Problem 15"

Revision as of 21:12, 25 November 2021

Problem

Isosceles triangle $ABC$ has $AB = AC = 3\sqrt6$ , and a circle with radius $5\sqrt2$ is tangent to line $AB$ at $B$ and to line $AC$ at $C$ . What is the area of the circle that passes through vertices $A$ , $B$ , and $C?$

$\textbf{(A) }24\pi\qquad\textbf{(B) }25\pi\qquad\textbf{(C) }26\pi\qquad\textbf{(D) }27\pi\qquad\textbf{(E) }28\pi$

Solution 1 (Similar Triangles)

$[asy] import olympiad; unitsize(50); pair A,B,C,D,E,I,F,G,O; A=origin; B=(2,3); C=(-2,3); D=(4.3,6.3); E=(-4.3,6.3); F=(1,1.5); G=(-1,1.5); O=circumcenter(A,B,C); // olympiad - circumcenter I=incenter(A,D,E); draw(A--B--C--cycle); dot(O); dot(I); dot(F); dot(G); draw(circumcircle(A,B,C)); // olympiad - circumcircle draw(incircle(A,D,E)); draw(I--B); draw(I--C); draw(I--A); draw(rightanglemark(A,C,I)); draw(rightanglemark(A,B,I)); draw(O--F); draw(O--G); draw(rightanglemark(A,F,O)); draw(rightanglemark(A,G,O)); label("$O$",O,W); label("$A$",A,S); label("$B$",B,N); label("$C$",C,W); label("$D$",F,S); label("$E$",G,W); label("$3\sqrt{6}$",(1.5,1.5),S); label("$3\sqrt{6}$",(-1.5,1.5),S); label("$5\sqrt{2}$",(1,3.625),N); label("$5\sqrt{2}$",(-1,3.625),N); label("$I$",I,N); label("$r$",(-0.25,1.5),E); label("$r$",(0.5,2.125),S); add(pathticks(A--F,1,0.5,0,2)); add(pathticks(F--B,1,0.5,0,2)); add(pathticks(A--G,1,0.5,0,2)); add(pathticks(G--C,1,0.5,0,2)); [/asy]$

Because circle $I$ is tangent to $\overline{AB}$ at $B, \angle{ABI} \cong 90^{\circ}$ . Because O is the circumcenter of $\bigtriangleup ABC, \overline{OD}$ is the perpendicular bisector of $\overline{AB}$ , and $\angle{BAI} \cong \angle{DAO}$ , so therefore $\bigtriangleup ADO \sim \bigtriangleup ABI$ by AA similarity. Then we have $\frac{AD}{AB} = \frac{DO}{BI} \implies \frac{1}{2} = \frac{r}{5\sqrt{2}} \implies r = \frac{5\sqrt{2}}{2}$ . We also know that $\overline{AD} = \frac{3\sqrt{6}}{2}$ because of the perpendicular bisector, so the hypotenuse of $\bigtriangleup ADO = \sqrt{(\frac{5\sqrt{2}}{2})^2+(\frac{3\sqrt{6}}{2})^2} = \sqrt{\frac{25}{2}+\frac{27}{2}} = \sqrt{26}$ . This is the radius of the circumcircle of $\bigtriangleup ABC$ , so the area of this circle is $26\pi = \boxed{C}$

Solution in Progress

~KingRavi

Solution 2 (Geometric Assumption)

Let the center of the first circle be $O.$ By Pythagorean Theorem, $AO=\sqrt{(3\sqrt{6})^2+(5\sqrt{2})^2}=2 \sqrt{26}$ Now, notice that since $\angle ABO$ is $90$ degrees, so arc $AO$ is $180$ degrees and $AO$ is the diameter. Thus, the radius is $\sqrt{26},$ so the area is $\boxed{26\pi}.$

- kante314

Video Solution by The Power of Logic

https://youtu.be/2lDDbOAmW18

~math2718281828459

Solution 3

Denote by $O$ the center of the circle that is tangent to line $AB$ at $B$ and to line $AC$ at $C$ .

Because this circle is tangent to line $AB$ at $B$ , $OB \perp AB$ and $OB = 5 \sqrt{2}$ . Because this circle is tangent to line $AC$ at $C$ , $OC \perp AC$ and $OC = 5 \sqrt{2}$ .

Because $AB = AC$ , $OB = OC$ , $AO = AO$ , $\triangle ABO \cong \triangle ACO$ . Hence, $\angle BAO = \angle CAO$ .

Let $AO$ and $BC$ meet at point $D$ . Because $AB = AC$ , $\angle BAO = \angle CAO$ , $AD = AD$ , $\triangle ABD \cong \triangle ACD$ .

Hence, $BD = CD$ and $\angle ADB = \angle ADC = 90^\circ$ .

Denote $\theta = \angle BAO$ . Hence, $\angle BAC = 2 \theta$ .

Denote by $R$ the circumradius of $\triangle ABC$ . In $\triangle ABC$ , following from the law of sines, $2 R = \frac{BC}{\sin \angle BAC}$ .

Therefore, the area of the circumcircle of $\triangle ABC$ is $\begin{align*} \pi R^2 & = \pi \left( \frac{BC}{2 \sin \angle BAC} \right)^2 \\ & = \pi \left( \frac{2 BD}{2 \sin \angle BAC} \right)^2 \\ & = \pi \left( \frac{BD}{\sin 2 \theta} \right)^2 \\ & = \pi \left( \frac{AB \sin \theta }{\sin 2 \theta} \right)^2 \\ & = \pi \left( \frac{AB \sin \theta }{2 \sin \theta \cos \theta} \right)^2 \\ & = \pi \left( \frac{AB }{2 \cos \theta} \right)^2 \\ & = \pi \left( \frac{AO}{2} \right)^2 \\ & = \frac{\pi}{4} \left( AB^2 + OB^2 \right) \\ & = 26 \pi . \end{align*}$

Therefore, the answer is $\boxed{\textbf{(C)} }26 \pi$ .

~Steven Chen (www.professorchenedu.com)

@@ Line 110: / Line 110: @@
 </cmath>
-Therefore, the answer is <math>\boxed{\textbf{(C) }}26 \pi</math>.
+Therefore, the answer is <math>\boxed{\textbf{(C)} }26 \pi</math>.
 ~Steven Chen (www.professorchenedu.com)

2021 Fall AMC 10A (Problems • Answer Key • Resources)
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