Point D is on side CB of triangle ABC. If ∠ CAD=∠ DAB=60°, AC=3 and AB=6, then the length of AD is

1983 AMC 12 Problem 19

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19.

Point

D

is on side

CB

of triangle

ABC.

\angle CAD=\angle DAB=60^\circ,

AC=3

and

AB=6,

then the length of

AD

2

2.5

3

3.5

4

The angle bisector theorem gives

\frac{DB}{CD}=\frac{AB}{AC}

Set

CD=t,

DB=2t,

and

AD=d;

then apply the law of cosines to the two triangles

By the angle bisector theorem, write

CD=t

and

DB=2t.

Let

AD=d.

The law of cosines in triangles

ACD

and

ABD,

whose angles at

A

are both

60^\circ,

gives

\begin{aligned} t^2&=9+d^2-3d,\\ 4t^2&=36+d^2-6d. \end{aligned}

Subtracting four times the first equation from the second yields

0=-3d^2+6d.

Since

d\gt0,

d=2.

Therefore, the correct answer is A.