Question

find the length of \\( \overline{pr} \\), if \\( \triangle pqr \\) is an isosceles with \\( pq \cong rq \\) and a perimeter of 23 cm.

(image of triangle pqr with sides: pq = 3x - 6, rq = 8x - 26, pr = 2x + 3)

multiple choice options: 6, 18, 4, 11

Explanation:

Step1: Set PQ = RQ

Since \( PQ \cong RQ \), set \( 3x - 6 = 8x - 26 \).
Solve: \( 3x - 6 = 8x - 26 \)
\( -6 + 26 = 8x - 3x \)
\( 20 = 5x \)
\( x = 4 \).

Step2: Find side lengths

• \( PQ = 3x - 6 = 3(4) - 6 = 6 \)
• \( RQ = 8x - 26 = 8(4) - 26 = 6 \)
• \( PR = 2x + 3 = 2(4) + 3 = 11 \)

Step3: Check perimeter

Perimeter: \( 6 + 6 + 11 = 23 \), matches. So \( PR = 11 \).

Answer:

11