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Difference between revisions of "2021 AMC 12A Problems/Problem 24"

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==Problem==
 
==Problem==
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Semicircle <math>\Gamma</math> has diameter <math>\overline{AB}</math> of length <math>14</math>. Circle <math>\Omega</math> lies tangent to <math>\overline{AB}</math> and intersects <math>\Gamma</math> at points <math>Q</math> and <math>R</math>. If <math>QR=3\sqrt{3}</math> and <math>\angle QPR = 60^{\circ}</math>, then the area of <math>\triangle PQR</math> equals <math>\tfrac{a\sqrt{b}}{c}</math>, where <math>a</math> and <math>c</math> are relatively prime positive integers, and <math>b</math> is a positive integer not divisible by the square of any prime. What is <math>a+b+c</math>?
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Semicircle <math>\Gamma</math> has diameter <math>\overline{AB}</math> of length <math>14</math>. Circle <math>\Omega</math> lies tangent to <math>\overline{AB}</math> at a point <math>P</math> and intersects <math>\Gamma</math> at points <math>Q</math> and <math>R</math>. If <math>QR=3\sqrt3</math> and <math>\angle QPR=60^\circ</math>, then the area of <math>\triangle PQR</math> equals <math>\tfrac{a\sqrt{b}}{c}</math>, where <math>a</math> and <math>c</math> are relatively prime positive integers, and <math>b</math> is a positive integer not divisible by the square of any prime. What is <math>a+b+c</math>?
  
 
<math>\textbf{(A) } 110\qquad\textbf{(B) } 114\qquad\textbf{(C) } 118\qquad\textbf{(D) } 122\qquad\textbf{(E) } 126\qquad</math>
 
<math>\textbf{(A) } 110\qquad\textbf{(B) } 114\qquad\textbf{(C) } 118\qquad\textbf{(D) } 122\qquad\textbf{(E) } 126\qquad</math>

Revision as of 12:52, 13 February 2021

Problem

Semicircle $\Gamma$ has diameter $\overline{AB}$ of length $14$. Circle $\Omega$ lies tangent to $\overline{AB}$ at a point $P$ and intersects $\Gamma$ at points $Q$ and $R$. If $QR=3\sqrt3$ and $\angle QPR=60^\circ$, then the area of $\triangle PQR$ equals $\tfrac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. What is $a+b+c$?

$\textbf{(A) } 110\qquad\textbf{(B) } 114\qquad\textbf{(C) } 118\qquad\textbf{(D) } 122\qquad\textbf{(E) } 126\qquad$

Video Solution by Punxsutawney Phil

https://youtube.com/watch?v=cEHF5iWMe9c

Video Solution by OmegaLearn (Similar Triangles, Law of Sines, Law of Cosines )

https://youtu.be/j965v6ahUZk

See also

2021 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 23 		Followed by
Problem 25
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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