Question

in the diagram below of triangle mno, p is the midpoint of \\(\overline{mo}\\) and q is the midpoint of \\(\overline{no}\\). if \\(pq = 8x - 8\\), and \\(mn = 2x + 40\\), what is the measure of \\(\overline{mn}\\)?

Explanation:

Step1: Apply Midline Theorem

In triangle \( MNO \), \( P \) is the midpoint of \( \overline{MO} \) and \( Q \) is the midpoint of \( \overline{NO} \). By the Midline Theorem (also known as the Midsegment Theorem), the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half as long. So, \( PQ=\frac{1}{2}MN \).

Step2: Set Up Equation

Substitute \( PQ = 8x - 8 \) and \( MN = 2x + 40 \) into the midline theorem relationship: \( 8x - 8=\frac{1}{2}(2x + 40) \).
Simplify the right - hand side: \( 8x - 8=x + 20 \).

Step3: Solve for \( x \)

Subtract \( x \) from both sides: \( 8x-x - 8=x - x+ 20 \), which gives \( 7x - 8 = 20 \).
Add 8 to both sides: \( 7x-8 + 8=20 + 8 \), so \( 7x=28 \).
Divide both sides by 7: \( x = \frac{28}{7}=4 \).

Step4: Find the length of \( \overline{MN} \)

Substitute \( x = 4 \) into the expression for \( MN \): \( MN=2x + 40 \).
\( MN=2\times4 + 40=8 + 40 = 48 \).

Answer:

The measure of \( \overline{MN} \) is \( 48 \).