In quadrilateral ABCD, diagonal AC divides the figure into two triangles, \triangle ABC an… — ACT

In quadrilateral $ABCD$ , diagonal $AC$ divides the figure into two triangles, $\triangle ABC$ and $\triangle ACD$ . It is given that $AB = 6$ , $BC = 10$ , $\angle ABC = 120^\circ$ , $\angle CAD = 45^\circ$ , and $\angle ADC = 60^\circ$ . What is the length of side $CD$ ?

$\frac{14\sqrt{6}}{3}$ Answer
B
$14$
C
$7\sqrt{3}$
D
$\frac{14\sqrt{2}}{3}$
E
$\sqrt{76}$

Answer

The length of side

CD

\frac{14\sqrt{6}}{3}

First, apply the Law of Cosines to

\triangle ABC

to find the length of diagonal

AC

AC^2 = 6^2 + 10^2 - 2(6)(10)\cos(120^\circ) = 36 + 100 - 120(-0.5) = 196

, which yields

AC = 14

. Next, use the Law of Sines in

\triangle ACD

\frac{CD}{\sin(45^\circ)} = \frac{14}{\sin(60^\circ)}

. Solving for

CD

gives

CD = 14 \cdot \frac{\sqrt{2}/2}{\sqrt{3}/2} = \frac{14\sqrt{2}}{\sqrt{3}} = \frac{14\sqrt{6}}{3}

Step-by-Step Solution

Apply the Law of Cosines in

\triangle ABC

to calculate the length of diagonal

AC

AC^2 = 6^2 + 10^2 - 2(6)(10)\cos(120^\circ) = 36 + 100 - 120\left(-\frac{1}{2}\right) = 136 + 60 = 196

, so

AC = 14

Two side lengths and the included angle of

\triangle ABC

are known.

Apply the Law of Sines in

\triangle ACD

to set up a proportion for side

CD

\frac{CD}{\sin(\angle CAD)} = \frac{AC}{\sin(\angle ADC)} \implies \frac{CD}{\sin(45^\circ)} = \frac{14}{\sin(60^\circ)}

The Law of Sines relates opposite sides and angles in

\triangle ACD

Solve for

CD

and rationalize the denominator.

CD = 14 \cdot \frac{\sin(45^\circ)}{\sin(60^\circ)} = 14 \cdot \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{3}}{2}} = \frac{14\sqrt{2}}{\sqrt{3}} = \frac{14\sqrt{6}}{3}

Evaluating the exact trigonometric values yields the final simplified length.

Key Concept

Law of Sines and Law of Cosines in Composite Triangles

Estimated Time:2m 0s

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