1953 AMC 12 Problem 28

Attempt Problem 28 of the 1953 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 1953 AMC 12 solutions, or check the answer key.

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28.

In triangle

ABC,

sides

a,

b

and

c

are opposite angles

A,

B

and

C

respectively.

\overline{AD}

bisects angle

A

and meets

\overline{BC}

at

D.

Then if

x=CD

and

y=BD

the correct proportion is:

\dfrac xa=\dfrac a{b+c}

\dfrac xb=\dfrac a{a+c}

\dfrac yc=\dfrac c{b+c}

\dfrac yc=\dfrac a{b+c}

\dfrac xy=\dfrac cb

Answer: D

Concepts:angle bisector theorem triangle proportion

Difficulty rating: 1470

Small hint:

Apply the angle bisector theorem to

\frac{BD}{DC}

Big hint:

Use

x+y=a

after writing

\frac{y}{c}=\frac{x}{b}

Solution:

The angle bisector theorem gives

\frac{y}{x}=\frac{c}{b},

or

\frac{y}{c}=\frac{x}{b}.

Since

x+y=a,

\frac yc=\frac xb=\frac{x+y}{b+c}=\frac a{b+c}.

Thus, the correct answer is D.