2022 AMC 10A Problem 13

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

Small hint:

Use the angle bisector theorem on triangle ABC

Big hint:

The perpendicular line creates an isosceles triangle before the final similarity

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.