Question

1. perpendicular/angle bisectors

a. find the length of pq in the given image below.
image of a triangle with points p, s, r on a vertical segment, q on a horizontal segment, pq labeled x + 2, rq labeled 2x - 1, and a right angle at s between the horizontal and vertical segments

Explanation:

Step1: Identify Perpendicular Bisector

The line \( SQ \) is a perpendicular bisector of \( PR \), so \( PQ = RQ \).

Step2: Set Up Equation

Since \( PQ = x + 2 \) and \( RQ = 2x - 1 \), set \( x + 2 = 2x - 1 \).

Step3: Solve for \( x \)

Subtract \( x \) from both sides: \( 2 = x - 1 \). Add 1 to both sides: \( x = 3 \).

Step4: Find \( PQ \) Length

Substitute \( x = 3 \) into \( PQ = x + 2 \): \( PQ = 3 + 2 = 5 \).

Answer:

The length of \( PQ \) is \( 5 \).