Question

In quadrilateral $ABCD,$ sides $\overline{AB}$ and $\overline{BC}$ both have length $10,$ sides $\overline{CD}$ and $\overline{DA}$ both have length $17,$ and the measure of angle $ADC$ is $60^\circ.$ What is the length of diagonal $\overline{AC}?$

$13.5$

$14$

$15.5$

$17$

$18.5$

Solution:

Note that $\triangle ADC$ is isosceles. This means that $\angle DAC = \angle DCA.$

We get that $\begin{gather*} \angle DAC + \angle DCA + \angle ADC = 180^{\circ} \\ \angle DAC + \angle DCA = 120^{\circ} \\ \angle DAC = \angle DCA = 60^{\circ}. \end{gather*}$

This shows that $\triangle ADC$ is equilateral. This gives us that $AC = CD = 17.$

Thus, D is the correct answer.

2005 AMC 8 Problem 9

Solution:

Problem 9 in Other Years