Question

in the diagram below of triangle jkl, m is the midpoint of \\( \overline{jl} \\) and n is the midpoint of \\( \overline{kl} \\). if \\( mn=-5x + 25 \\), and \\( jk = 35 + 5x \\), what is the measure of \\( \overline{mn} \\)?

Explanation:

Step1: Use 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. So, \(MN=\frac{1}{2}JK\).

Step2: Substitute the given expressions

Given \(MN = - 5x+25\) and \(JK = 35 + 5x\), we substitute into the equation \(MN=\frac{1}{2}JK\). So, \(-5x + 25=\frac{1}{2}(35 + 5x)\).

Step3: Solve the equation for \(x\)

Multiply both sides of the equation \(-5x + 25=\frac{1}{2}(35 + 5x)\) by \(2\) to get \(2(-5x + 25)=35 + 5x\).
Expand the left - hand side: \(-10x+50 = 35+5x\).
Add \(10x\) to both sides: \(50=35 + 15x\).
Subtract \(35\) from both sides: \(15 = 15x\).
Divide both sides by \(15\): \(x = 1\).

Step4: Find the length of \(MN\)

Substitute \(x = 1\) into the expression for \(MN\). Since \(MN=-5x + 25\), then \(MN=-5(1)+25\).
Calculate: \(MN=-5 + 25\).

Answer:

\(20\)