Question

Let $\triangle ABC$ be a scalene triangle. Point $P$ lies on $\overline{BC}$ so that $\overline{AP}$ bisects $\angle BAC.$ The line through $B$ perpendicular to $\overline{AP}$ intersects the line through $A$ parallel to $\overline{BC}$ at point $D.$ Suppose $BP = 2$ and $PC = 3.$ What is $AD?$

$8$

$9$

$10$

$11$

$12$

Solution:

Consider the following diagram:

By the Angle Bisector Theorem, we can label $AB$ as $2x$ and $AY$ as $3x.$

We also get that $\triangle ABY$ is isosceles since $AP \perp BY.$ Therefore, $AY = AB = 2x.$

Since $AD$ and $BC$ are parallel, we know that $\triangle BYC \sim \triangle DYA.$

$YC = AC - AY = 3x - 2x = x,$ so $AY = 2YC.$

Using similar triangles, we get that $AD = 2BC = 10.$

Thus, C is the correct answer.

2022 AMC 10A Problem 13

Solution:

Problem 13 in Other Years