Question

In $\triangle ABC$ we have $AB = 7,$ $AC = 8,$ and $BC = 9.$ Point $D$ is on the circumscribed circle of the triangle so that $\overline{AD}$ bisects $\angle BAC.$ What is the value of $AD/CD?$

$\dfrac{9}{8}$

$\dfrac{5}{3}$

$2$

$\dfrac{17}{7}$

$\dfrac{5}{2}$

Answer: B

Concepts:inscribed angle similarity angle bisector theorem

Difficulty rating: 2030

Solution:

Let $\overline{AD}$ meet $\overline{BC}$ at $E.$ Since $\angle ABC$ and $\angle ADC$ subtend the same arc, they are equal, and $\angle EAB = \angle CAD,$ so $\triangle ABE \sim \triangle ADC.$

Hence $\dfrac{AD}{CD} = \dfrac{AB}{BE}.$

By the Angle Bisector Theorem, $\dfrac{BE}{EC} = \dfrac{AB}{AC},$ so $BE = \dfrac{AB \cdot BC}{AB + AC} = \dfrac{7 \cdot 9}{15}.$

Therefore $\begin{aligned} \dfrac{AD}{CD} &= \dfrac{AB}{BE} = \dfrac{AB + AC}{BC} \\ &= \dfrac{15}{9} = \dfrac{5}{3}. \end{aligned}$

Thus, the correct answer is B.

2004 AMC 10B Problem 24

Answer: B

Difficulty rating: 2030

Solution:

Problem 24 in Other Years