Question

in the diagram below of triangle nop, q is the midpoint of \\( \overline { n p } \\) and r is the midpoint of \\( \overline { o p } \\). if qr = 48 - 5x, and no = -16 + 6x, what is the measure of \\( \overline { q r } \\)?

Explanation:

Step1: Use the mid - segment theorem

The mid - segment theorem states that the length of the mid - segment of a triangle (a segment connecting the midpoints of two sides of a triangle) is half the length of the third side.
Since \(Q\) is the midpoint of \(\overline{NP}\) and \(R\) is the midpoint of \(\overline{OP}\), then \(QR=\frac{1}{2}NO\).
So, \(48 - 5x=\frac{1}{2}(-16 + 6x)\).

Step2: Solve the equation for \(x\)

Multiply both sides of the equation \(48 - 5x=\frac{1}{2}(-16 + 6x)\) by \(2\) to get rid of the fraction:
\(2(48 - 5x)=-16 + 6x\).
Expand the left - hand side: \(96-10x=-16 + 6x\).
Add \(10x\) to both sides: \(96=-16 + 6x+10x\).
Combine like terms: \(96=-16 + 16x\).
Add \(16\) to both sides: \(96 + 16=16x\), so \(112 = 16x\).
Divide both sides by \(16\): \(x = 7\).

Step3: Find the length of \(QR\)

Substitute \(x = 7\) into the expression for \(QR\).
Since \(QR=48-5x\), then \(QR=48-5\times7\).
\(QR=48 - 35\).

Answer:

\(13\)