In right triangle ABC, the right angle is located at vertex B, and the measure of angle A … — ACT

In right triangle $ABC$ , the right angle is located at vertex $B$ , and the measure of angle $A$ is $30^\circ$ . The hypotenuse $AC$ has a length of 16 inches. Segment $BD$ is an altitude drawn from vertex $B$ perpendicular to hypotenuse $AC$ at point $D$ . What is the length, in inches, of segment $CD$ ?

Answer: 4 inches

Answer

The length of segment CD is 4 inches.

In right triangle ABC with angle A = 30°, the side opposite angle A (BC) is half the hypotenuse AC, so BC = 16 / 2 = 8 inches. Drawing altitude BD creates smaller right triangle BCD with right angle at D and angle C = 60°. This makes triangle BCD another 30°-60°-90° right triangle where segment BC = 8 inches is the hypotenuse. Segment CD lies opposite the 30° angle DBC, meaning CD is half of BC: 8 / 2 = 4 inches.

Step-by-Step Solution

Determine the length of leg BC in right triangle ABC

BC = 8 inches

In a 30°-60°-90° triangle, the length of the side opposite the 30° angle is equal to half the length of the hypotenuse. Since hypotenuse AC = 16 inches, BC = 16 / 2 = 8 inches.

Identify the angles of right triangle BCD

Angle C = 60°, Angle BDC = 90°, and Angle DBC = 30°

Since angle A = 30° in right triangle ABC, angle C must equal 90° - 30° = 60°. Altitude BD creates right angle BDC = 90°, leaving angle DBC = 180° - 90° - 60° = 30°.

Calculate the length of segment CD in 30°-60°-90° triangle BCD

CD = 4 inches

In triangle BCD, segment BC (8 inches) is the hypotenuse. Segment CD lies opposite the 30° angle DBC, so its length is half of the hypotenuse BC: 8 / 2 = 4 inches.

Key Concept

Altitude to Hypotenuse in Special 30°-60°-90° Right Triangles

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