Question

q is the midpoint of \\(\overline{pr}\\). if \\(pq = 8x - 5\\) and \\(pr = 15x - 8\\), what is \\(pq\\)? simplify your answer and write it as a proper fraction, mixed number, or integer.

Explanation:

Step1: Use midpoint property

Since Q is the midpoint of \( \overline{PR} \), \( PQ=\frac{1}{2}PR \). So we have the equation \( 8x - 5=\frac{1}{2}(15x - 8) \).

Step2: Solve for x

Multiply both sides of the equation by 2 to eliminate the fraction: \( 2(8x - 5)=15x - 8 \).
Expand the left side: \( 16x-10 = 15x - 8 \).
Subtract \( 15x \) from both sides: \( 16x-15x-10=15x - 15x- 8 \), which simplifies to \( x - 10=-8 \).
Add 10 to both sides: \( x-10 + 10=-8 + 10 \), so \( x = 2 \).

Step3: Find PQ

Substitute \( x = 2 \) into the expression for PQ: \( PQ=8x - 5=8\times2-5=16 - 5 = 11 \).

Answer:

11