Difference between revisions of "2023 AMC 12B Problems/Problem 3"
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==Solution 1== | ==Solution 1== | ||
| − | Because the triangle are right triangles, we know the hypotenuses are diameters of circles <math>A</math> and <math>B</math>. Thus, their radii are 2.5 and 6.5 (respectively). Square the two numbers and multiply <math>\pi</math> to get <math>6.25\pi</math> and <math>42.25\pi</math> as the areas of the circles. Multiply 4 on both numbers to get <math>25\pi</math> and <math>169\pi</math>. Cancel out the <math>\pi</math>, and lastly, divide, to get your answer | + | Because the triangle are right triangles, we know the hypotenuses are diameters of circles <math>A</math> and <math>B</math>. Thus, their radii are 2.5 and 6.5 (respectively). Square the two numbers and multiply <math>\pi</math> to get <math>6.25\pi</math> and <math>42.25\pi</math> as the areas of the circles. Multiply 4 on both numbers to get <math>25\pi</math> and <math>169\pi</math>. Cancel out the <math>\pi</math>, and lastly, divide, to get your answer <math>\boxed{\textbf(D)\frac{25}{169}}</math>. |
~Failure.net | ~Failure.net | ||
Revision as of 13:29, 15 November 2023
Problem
A 3-4-5 right triangle is inscribed circle 
, and a 5-12-13 right triangle is inscribed in circle 
. What is the ratio of the area of circle to circle ?
Solution 1
Because the triangle are right triangles, we know the hypotenuses are diameters of circles and . Thus, their radii are 2.5 and 6.5 (respectively). Square the two numbers and multiply 
to get 
and 
as the areas of the circles. Multiply 4 on both numbers to get 
and 
. Cancel out the , and lastly, divide, to get your answer 
.
~Failure.net
