2021 Fall AMC 12B Problems/Problem 22
Problem
Right triangle has side lengths , , and .
A circle centered at is tangent to line at and passes through . A circle centered at is tangent to line at and passes through . What is ?
Solution 1 (Analytic Geometry)
In a Cartesian plane, let and be respectively.
By analyzing the behaviors of the two circles, we set be and be .
Hence derive the two equations:
Considering the coordinates of and for the two equations respectively, we get:
Solve to get and
Through using the distance formula,
$OP=\sqrt{(8-\frac{25}{4})^2+(\frac{25}{3}-6)^2}=\boxed{\textbf{(C)\ \frac{35}{12}}}$ (Error compiling LaTeX. Unknown error_msg).
~Wilhelm Z
2021 Fall AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 21 |
Followed by Problem 23 |
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