Difference between revisions of "2017 AMC 8 Problems/Problem 22"
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<math>\textbf{(A) }\frac{7}{6}\qquad\textbf{(B) }\frac{13}{5}\qquad\textbf{(C) }\frac{59}{18}\qquad\textbf{(D) }\frac{10}{3}\qquad\textbf{(E) }\frac{60}{13}</math> | <math>\textbf{(A) }\frac{7}{6}\qquad\textbf{(B) }\frac{13}{5}\qquad\textbf{(C) }\frac{59}{18}\qquad\textbf{(D) }\frac{10}{3}\qquad\textbf{(E) }\frac{60}{13}</math> | ||
| − | ==Solution | + | ==Solution== |
We can reflect triangle <math>ABC</math> on line <math>AC.</math> This forms the triangle <math>AB'C</math> and a circle out of the semicircle. Let us call the center of the circle <math>O.</math> | We can reflect triangle <math>ABC</math> on line <math>AC.</math> This forms the triangle <math>AB'C</math> and a circle out of the semicircle. Let us call the center of the circle <math>O.</math> | ||
We can see that Circle <math>O</math> is the incircle of <math>AB'C.</math> We can use the formula for finding the radius of the incircle to solve this problem. The are of <math>AB'C</math> is <math>12\times5 = 60.</math> The semiperimeter is <math>5+13 = 18.</math> Simplifying <math>\dfrac{60}{18} = \dfrac{10}{3}.</math> Our answer is therefore <math>\text{D)}</math> <math>\dfrac{10}{3}.</math> | We can see that Circle <math>O</math> is the incircle of <math>AB'C.</math> We can use the formula for finding the radius of the incircle to solve this problem. The are of <math>AB'C</math> is <math>12\times5 = 60.</math> The semiperimeter is <math>5+13 = 18.</math> Simplifying <math>\dfrac{60}{18} = \dfrac{10}{3}.</math> Our answer is therefore <math>\text{D)}</math> <math>\dfrac{10}{3}.</math> | ||
Revision as of 14:40, 22 November 2017
Problem 22
In the right triangle 
, 
, 
, and angle 
is a right angle. A semicircle is inscribed in the triangle as shown. What is the radius of the semicircle?
![[asy] draw((0,0)--(12,0)--(12,5)--(0,0)); draw(arc((8.67,0),(12,0),(5.33,0))); label("$A$", (0,0), W); label("$C$", (12,0), E); label("$B$", (12,5), NE); label("$12$", (6, 0), S); label("$5$", (12, 2.5), E);[/asy]](http://latex.artofproblemsolving.com/1/7/f/17f66d4cef8b689eb626ec8c226380e1e72d8719.png)
Solution
We can reflect triangle on line 
This forms the triangle 
and a circle out of the semicircle. Let us call the center of the circle 
We can see that Circle 
is the incircle of 
We can use the formula for finding the radius of the incircle to solve this problem. The are of is 
The semiperimeter is 
Simplifying 
Our answer is therefore 
