In the figure below, quadrilateral ABCD is composed of two right triangles, \triangle ABC … — ACT

In the figure below, quadrilateral $ABCD$ is composed of two right triangles, $\triangle ABC$ and $\triangle ACD$ . The measure of $\angle ABC$ is $90^\circ$ , and the measure of $\angle ACD$ is $90^\circ$ . The side lengths are $AB = 3$ units and $BC = 4$ units. If the measure of $\angle CAD$ is $60^\circ$ , what is the length, in units, of segment $CD$ ?

A
$\sqrt{21}$
$5\sqrt{3}$ Answer
C
$7\sqrt{3}$
D
10
E
$\sqrt{39}$

Answer

The length of segment

CD

5\sqrt{3}

units.

The correct answer is the length of

5\sqrt{3}

units. By using the Pythagorean theorem on the first right triangle

\triangle ABC

, the length of the hypotenuse is

AC = \sqrt{3^2 + 4^2} = 5

. Since

\triangle ACD

is a 30-60-90 right triangle with a right angle at

C

and

\angle CAD = 60^\circ

, the side

AC

is the shorter leg (opposite the

30^\circ

angle). The length of the longer leg

CD

(opposite the

60^\circ

angle) is therefore

AC\sqrt{3} = 5\sqrt{3}

Step-by-Step Solution

Use the Pythagorean theorem in right triangle

\triangle ABC

to find the length of the hypotenuse

AC

AC = 5

Since

\triangle ABC

is a right triangle with legs

AB = 3

and

BC = 4

, the hypotenuse is

AC = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5

Identify the type of right triangle for

\triangle ACD

\triangle ACD

is a 30-60-90 right triangle.

The triangle has a right angle at

C

(measure of

\angle ACD = 90^\circ

) and an acute angle at

A

(measure of

\angle CAD = 60^\circ

), which leaves the remaining angle

\angle ADC = 30^\circ

Apply the special right triangle ratios to find the length of leg

CD

CD = 5\sqrt{3}

In a 30-60-90 triangle, the leg opposite the

60^\circ

angle is

\sqrt{3}

times the leg opposite the

30^\circ

angle. Here,

AC = 5

is opposite the

30^\circ

angle, so the longer leg

CD = AC\sqrt{3} = 5\sqrt{3}

Key Concept

Using the Pythagorean theorem to find a shared side and then applying special right triangle ratios (30-60-90) to solve for an unknown length.

Alternative Method

Instead of using the special right triangle ratios directly, right triangle trigonometry can be applied:

\tan(60^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{CD}{AC}

. Since

\tan(60^\circ) = \sqrt{3}

and

AC = 5

, we have

\sqrt{3} = \frac{CD}{5}

, which yields

CD = 5\sqrt{3}

Estimated Time:1m 0s

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Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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