Question

in the diagram below of triangle opq, r is the midpoint of \overline{oq} and s is the midpoint of \overline{pq}. if rs = 8x - 66, and op = -4x + 48, what is the measure of \overline{op}?

Explanation:

Step1: Use the mid - segment theorem

In \(\triangle OPQ\), since \(R\) is the mid - point of \(\overline{OQ}\) and \(S\) is the mid - point of \(\overline{PQ}\), by the mid - segment theorem, \(RS=\frac{1}{2}OP\).

Step2: Substitute the given expressions

Given \(RS = 8x-66\) and \(OP=-4x + 48\). Substitute into \(RS=\frac{1}{2}OP\), we get \(8x-66=\frac{1}{2}(-4x + 48)\).

Step3: Solve the equation for \(x\)

Multiply both sides of the equation \(8x-66=\frac{1}{2}(-4x + 48)\) by \(2\) to get \(16x-132=-4x + 48\).
Add \(4x\) to both sides: \(16x+4x-132=-4x + 4x+ 48\), which simplifies to \(20x-132 = 48\).
Add \(132\) to both sides: \(20x-132+132=48+132\), so \(20x=180\).
Divide both sides by \(20\): \(x=\frac{180}{20}=9\).

Step4: Find the length of \(OP\)

Substitute \(x = 9\) into \(OP=-4x + 48\). Then \(OP=-4\times9+48\).
\(OP=-36 + 48\).
\(OP = 24\).

Answer:

\(24\)