In right triangle PQR, the measure of \angle PQR is 90^\circ. Segment QS is an altitude dr… — ACT

In right triangle $PQR$ , the measure of $\angle PQR$ is $90^\circ$ . Segment $QS$ is an altitude drawn perpendicular to hypotenuse $PR$ with point $S$ lying on $PR$ . If $PQ = 15$ centimeters and $PS = 9$ centimeters, what is the length, in centimeters, of hypotenuse $PR$ ?

Answer: 25 cm

Answer

The length of hypotenuse

PR

is 25 centimeters.

Using the leg-hypotenuse geometric mean theorem for right triangles (

PQ^2 = PS \cdot PR

), substituting

PQ = 15

and

PS = 9

yields

15^2 = 9 \cdot PR \implies 225 = 9 \cdot PR

, solving directly to

PR = 25

cm.

Step-by-Step Solution

Calculate the length of altitude

QS

using the Pythagorean theorem on right triangle

\triangle PQS

QS = \sqrt{15^2 - 9^2} = \sqrt{225 - 81} = \sqrt{144} = 12

Altitude

QS

forms right angle

\angle PSQ = 90^\circ

, making

\triangle PQS

a right triangle.

Determine the length of hypotenuse segment

SR

using the geometric mean relationship

QS^2 = PS \cdot SR

12^2 = 9 \cdot SR \implies 144 = 9 \cdot SR \implies SR = 16

The altitude to the hypotenuse divides the original right triangle into two smaller similar right triangles.

Sum the segment lengths

PS

and

SR

to find the total length of hypotenuse

PR

PR = PS + SR = 9 + 16 = 25

Point

S

lies directly on segment

PR

between endpoints

P

and

R

Key Concept

Geometric Mean Theorem and Pythagorean Theorem in Right Triangles

Estimated Time:1m 30s

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