Question

in the diagram below of triangle lmn, o is the midpoint of ln and p is the midpoint of mn. if op = -7 + 2x, and lm = -2x + 40, what is the measure of op? answer op =

Explanation:

Step1: Use the mid - segment theorem

The mid - segment theorem states that the length of the mid - segment (OP) of a triangle is half the length of the third side (LM). So, \(OP=\frac{1}{2}LM\).
Given \(OP = 2x-7\) and \(LM=-2x + 40\), we substitute into the formula: \(2x-7=\frac{1}{2}(-2x + 40)\).

Step2: Solve the equation for \(x\)

Multiply both sides of the equation \(2x-7=\frac{1}{2}(-2x + 40)\) by \(2\) to get rid of the fraction:

$$
LATEXBLOCK0
$$

Add \(2x\) to both sides: \(4x+2x-14=-2x+2x + 40\), which simplifies to \(6x-14 = 40\).
Add \(14\) to both sides: \(6x-14+14=40+14\), so \(6x=54\).
Divide both sides by \(6\): \(x = 9\).

Step3: Find the length of \(OP\)

Substitute \(x = 9\) into the formula for \(OP\): \(OP=2x-7\).
\(OP=2\times9-7\).
\(OP = 18-7\).
\(OP=11\).

Answer:

\(11\)