In right triangle ABC, the measure of \angle B is 90^\circ, the measure of \angle A is 60^… — ACT

In right triangle $ABC$ , the measure of $\angle B$ is $90^\circ$ , the measure of $\angle A$ is $60^\circ$ , and the length of $AC$ is $16$ units. Point $D$ lies on side $BC$ such that the measure of $\angle ADB$ is $45^\circ$ . What is the length of segment $CD$ , rounded to the nearest tenth?

Answer: 5.9 units

Answer

The length of segment CD is approximately 5.9 units.

By recognizing that triangle ABC is a 30-60-90 right triangle, the shorter leg AB is found to be 8 (half of the hypotenuse 16), and the longer leg BC is 8*sqrt(3). Because triangle ABD is a 45-45-90 right triangle, the leg BD is equal to the leg AB, which is 8. Subtracting BD from BC yields CD = 8*sqrt(3) - 8, which is approximately 5.9.

Step-by-Step Solution

Determine the type of triangle ABC

Triangle ABC is a 30-60-90 special right triangle.

The triangle has a right angle (90 degrees) at B and an angle of 60 degrees at A, which leaves 30 degrees for angle C.

Calculate the lengths of sides AB and BC

AB = 8 units and BC = 8*sqrt(3) units.

In a 30-60-90 triangle with hypotenuse AC = 16, the side opposite 30 degrees (AB) is half the hypotenuse, and the side opposite 60 degrees (BC) is the shorter leg multiplied by sqrt(3).

Determine the type of triangle ABD

Triangle ABD is a 45-45-90 special right triangle.

Since D lies on BC, angle ABD is a right angle (90 degrees). Given that angle ADB is 45 degrees, the remaining angle BAD must also be 45 degrees.

Calculate the length of side BD

BD = 8 units.

In a 45-45-90 right triangle, the two legs opposite the 45-degree angles are equal in length, so BD = AB.

Calculate the length of segment CD and round to the nearest tenth

CD ≈ 5.9 units.

Since D lies on side BC, CD = BC - BD = 8*sqrt(3) - 8 ≈ 8(1.732) - 8 = 13.856 - 8 = 5.856, which rounds to 5.9.

Key Concept

Applying properties of 30-60-90 and 45-45-90 special right triangles to find segment lengths within nested figures.

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