Question

in the diagram below of triangle lmn, o is the midpoint of \\( \overline { l n } \\) and p is the midpoint of \\( \overline { m n } \\). if \\( o p = - 36 + 7 x \\), and \\( l m = 56 - 2 x \\), what is the measure of \\( \overline { l m } \\)?

Explanation:

Step1: Apply the mid - segment theorem

The mid - segment theorem states that the length of the mid - segment (a segment connecting the midpoints of two sides of a triangle) is half the length of the third side. Here, \(OP\) is the mid - segment and \(LM\) is the third side. So, \(OP=\frac{1}{2}LM\).
Given \(OP = - 36+7x\) and \(LM = 56 - 2x\), we substitute into the formula: \(-36 + 7x=\frac{1}{2}(56 - 2x)\).

Step2: Solve the equation for \(x\)

Multiply both sides of the equation \(-36 + 7x=\frac{1}{2}(56 - 2x)\) by \(2\) to get rid of the fraction:

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Add \(2x\) to both sides: \(-72 + 14x+2x=56-2x + 2x\), which simplifies to \(-72+16x=56\).
Add \(72\) to both sides: \(-72 + 72+16x=56 + 72\), so \(16x=128\).
Divide both sides by \(16\): \(x=\frac{128}{16}=8\).

Step3: Find the length of \(LM\)

Substitute \(x = 8\) into the formula for \(LM\). Since \(LM = 56-2x\), then \(LM=56-2\times8\).

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Answer:

\(40\)