Question

point q is on line segment $overline{pr}$. given $qr = 3x, pq=x + 6$, and $pr = 5x-4$, determine the numerical length of $overline{pq}$.

Explanation:

Step1: Use segment - addition postulate

Since point Q is on line segment $\overline{PR}$, we know that $PQ + QR=PR$. Substitute the given expressions: $(x + 6)+3x=5x - 4$.

Step2: Simplify the left - hand side of the equation

Combine like terms: $x+3x + 6=5x - 4$, which simplifies to $4x+6=5x - 4$.

Step3: Solve for x

Subtract $4x$ from both sides: $4x+6-4x=5x - 4-4x$, getting $6=x - 4$. Then add 4 to both sides: $6 + 4=x-4 + 4$, so $x = 10$.

Step4: Find the length of $\overline{PQ}$

Substitute $x = 10$ into the expression for $PQ$: $PQ=x + 6$. Then $PQ=10 + 6=16$.

Answer:

16