In trapezoid ABCD, side AB is perpendicular to parallel bases AD and BC. Diagonal AC is pe… — GRE General Test

In trapezoid $ABCD$ , side $AB$ is perpendicular to parallel bases $AD$ and $BC$ . Diagonal $AC$ is perpendicular to side $CD$ . If the measure of angle $CAD$ is $30^\circ$ and the length of $BC$ is $9$ , what is the length of side $CD$ ?

Answer: 6

Answer

Because base

BC

and base

AD

are parallel, transversal

AC

creates equal alternate interior angles, making

\angle ACB = \angle CAD = 30^\circ

. Triangle

ABC

is a right triangle with

\angle B = 90^\circ

, so

BC

is adjacent to

30^\circ

. The ratio of the side adjacent to

30^\circ

to the hypotenuse

AC

\frac{\sqrt{3}}{2}

, which yields

AC = \frac{9}{\sqrt{3}/2} = 6\sqrt{3}

. Triangle

ACD

is also a

30^\circ-60^\circ-90^\circ

right triangle with right angle at

C

. Side

CD

is opposite the

30^\circ

angle and side

AC

is opposite the

60^\circ

angle. The ratio of the short leg to the long leg is

\frac{1}{\sqrt{3}}

, giving

CD = \frac{6\sqrt{3}}{\sqrt{3}} = 6

Step-by-Step Solution

Determine angle measures in right triangle ABC using parallel line properties

Angle ACB = 30 degrees and angle BAC = 60 degrees

Since base BC is parallel to base AD, alternate interior angles formed by transversal AC are equal: angle ACB = angle CAD = 30 degrees.

Calculate the length of diagonal AC using special right triangle ratios

AC = 6*sqrt(3)

In 30-60-90 right triangle ABC, the ratio of the side adjacent to 30 degrees (BC) to the hypotenuse (AC) is sqrt(3)/2. Therefore, 9 / AC = sqrt(3)/2, which gives AC = 18 / sqrt(3) = 6*sqrt(3).

Calculate the length of side CD using right triangle ACD

CD = 6

In 30-60-90 right triangle ACD, angle ACD = 90 degrees and angle CAD = 30 degrees. The ratio of the short leg opposite 30 degrees (CD) to the long leg opposite 60 degrees (AC) is 1/sqrt(3). Therefore, CD = AC / sqrt(3) = (6*sqrt(3)) / sqrt(3) = 6.

Key Concept

Side ratios of 30-60-90 special right triangles and alternate interior angles

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