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 $AD?$

$8$

$9$

$10$

$11$

$12$

Answer: C

Concepts:angle bisector theorem similarity isosceles triangle

Difficulty rating: 1540

Solution:

Consider the following diagram:

Let $Y$ be the intersection of $\overline{BD}$ and $\overline{AC}.$ By the Angle Bisector Theorem, $AB:AC=BP:PC=2:3,$ so write $AB=2x$ and $AC=3x.$

Reflection across the angle bisector $\overline{AP}$ sends ray $AB$ to ray $AC.$ Because $BY\perp AP,$ it sends $B$ to $Y.$ Thus $AY=AB=2x,$ and hence $YC=AC-AY=x.$

Since $AD\parallel BC,$ with $B,Y,D$ collinear and $A,Y,C$ collinear, we have $\triangle BYC\sim\triangle DYA.$ Therefore $\frac{AD}{BC}=\frac{AY}{YC}=2.$

Finally, $BC=BP+PC=5,$ so $AD=2BC=10.$

Thus, C is the correct answer.

2022 AMC 10A Problem 13

Answer: C

Difficulty rating: 1540

Solution:

Problem 13 in Other Years