1961 AMC 12 Problem 31

Attempt Problem 31 of the 1961 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1961 AMC 12 solutions, or check the answer key.

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31.

In triangle

ABC

the ratio

AC:CB

is

3:4.

The bisector of the exterior angle at

C

intersects

BA

extended at

P

(

A

is between

P

and

B

). The ratio

PA:AB

is:

1:3

3:4

4:3

3:1

7:1

Answer: D

Concepts:angle bisector theorem ratio and proportion

Difficulty rating: 1300

Small hint:

Apply the exterior angle bisector theorem to the full lengths

PA

and

PB

Big hint:

Use

PB=PA+AB

Solution:

The exterior angle bisector theorem gives

\frac{PA}{PB}=\frac{AC}{CB}=\frac34.

Since

PB=PA+AB,

write

PA=3t,

PB=4t.

Then

AB=t,

so

PA:AB=3:1.

Thus, the correct answer is D.