In right triangle ABC, the right angle is located at vertex B. Point D lies on leg BC such… — ACT

In right triangle $ABC$ , the right angle is located at vertex $B$ . Point $D$ lies on leg $BC$ such that $AB = 12$ inches and $BD = 9$ inches. If segment $AD$ is equal in length to segment $DC$ , what is the length of leg $BC$ , in inches?

Answer: 24 inches

Answer

The length of leg

BC

is 24 inches.

Applying the Pythagorean theorem to right triangle

ABD

gives

AD = \sqrt{12^2 + 9^2} = 15

inches. Because segment

AD

equals segment

DC

DC

is also 15 inches. Adding the lengths of segments

BD

and

DC

gives

BC = 9 + 15 = 24

inches.

Step-by-Step Solution

Calculate the length of hypotenuse

AD

in right triangle

ABD

AD = 15

inches

Apply the Pythagorean theorem:

AD = \sqrt{AB^2 + BD^2} = \sqrt{12^2 + 9^2} = 15

Determine the length of segment

DC

DC = 15

inches

It is given that segment

AD

is equal in length to segment

DC

Calculate the total length of leg

BC

BC = 24

inches

Add the adjacent segment lengths along leg

BC

BC = BD + DC = 9 + 15 = 24

Key Concept

Applying the Pythagorean theorem to adjacent right triangles within geometric figures.

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