In right triangle PQR, the measure of angle PQR is 90^\circ. Point S lies on side PQ. If t… — SAT

In right triangle $PQR$ , the measure of angle $PQR$ is $90^\circ$ . Point $S$ lies on side $PQ$ . If the length of side $QR$ is $15$ , the length of segment $SR$ is $17$ , and the length of segment $PS$ is $12$ , what is the length of side $PR$ ?

Cevap: 25

Cevap

The length of side

PR

25

First, the Pythagorean theorem is applied to the smaller right triangle

SQR

to solve for the unknown leg

QS

QS = \sqrt{SR^2 - QR^2} = \sqrt{17^2 - 15^2} = 8

. Next, the total length of leg

PQ

is the sum of its parts,

PS

and

QS

, which gives

PQ = 12 + 8 = 20

. Finally, the Pythagorean theorem is applied to the main right triangle

PQR

to find the hypotenuse

PR

PR = \sqrt{PQ^2 + QR^2} = \sqrt{20^2 + 15^2} = \sqrt{400 + 225} = \sqrt{625} = 25

Adım Adım Çözüm

Use the Pythagorean theorem on the right triangle

SQR

to find the length of segment

QS

QS = 8

Triangle

SQR

shares the right angle at

Q

with triangle

PQR

. Thus,

QS^2 + QR^2 = SR^2 \implies QS^2 + 15^2 = 17^2 \implies QS = 8

Determine the length of side

PQ

PQ = 20

Since

S

lies on side

PQ

, the length of

PQ

is the sum of

PS

and

QS

PQ = 12 + 8 = 20

Use the Pythagorean theorem on right triangle

PQR

to calculate the length of the hypotenuse

PR

PR = 25

PQ^2 + QR^2 = PR^2 \implies 20^2 + 15^2 = PR^2 \implies 400 + 225 = 625 \implies PR = 25

Anahtar Kavram

Multi-step application of the Pythagorean theorem in nested right triangles

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