In the geometric configuration formed by adjacent triangles ABC and ACD sharing segment AC… — GRE General Test

In the geometric configuration formed by adjacent triangles $ABC$ and $ACD$ sharing segment $AC$ , $\angle ABC = 90^\circ$ and $\angle ACD = 90^\circ$ . The lengths of the sides of triangle $ABC$ are $AB = 9$ and $BC = 12$ . If $\angle ADC = 30^\circ$ , what is the length of segment $AD$ ?

Answer: 30

Answer

Applying the Pythagorean theorem to right triangle ABC gives AC = sqrt(9^2 + 12^2) = 15. In right triangle ACD with angle ADC = 30 degrees, side AC is opposite the 30-degree angle. By the 30-60-90 triangle side ratio (1 : sqrt(3) : 2), the hypotenuse AD is equal to twice the short leg AC, giving AD = 2 * 15 = 30.

Step-by-Step Solution

Calculate hypotenuse AC of right triangle ABC using the Pythagorean theorem.

AC = 15

In right triangle ABC with legs 9 and 12, AC^2 = 9^2 + 12^2 = 225, so AC = 15.

Identify the side relationships in special right triangle ACD.

AC is the shorter leg opposite the 30-degree angle ADC, and AD is the hypotenuse.

Since angle ACD is 90 degrees and angle ADC is 30 degrees, triangle ACD is a 30-60-90 right triangle.

Compute the hypotenuse AD from short leg AC.

AD = 30

In any 30-60-90 right triangle, the hypotenuse is twice the length of the leg opposite the 30-degree angle.

Key Concept

Pythagorean Theorem and Special Right Triangles

Estimated Time:1m 30s

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