Question

in the diagram below of triangle klm, n is the midpoint of \\( \overline{km} \\) and o is the midpoint of lm. if \\( no = 84 - 7x \\), and \\( kl = -3x + 69 \\), what is the measure of \\( \overline{no} \\)?

Explanation:

Step1: Apply the mid - segment theorem

In a triangle, the mid - segment (a segment connecting the midpoints of two sides) is parallel to the third side and half its length. So, \(NO=\frac{1}{2}KL\).

Step2: Substitute the given expressions

Given \(NO = 84-7x\) and \(KL=-3x + 69\), we substitute into the equation \(NO=\frac{1}{2}KL\). So, \(84-7x=\frac{1}{2}(-3x + 69)\).

Step3: Solve the equation for \(x\)

Multiply both sides of the equation \(84-7x=\frac{1}{2}(-3x + 69)\) by 2 to get rid of the fraction:
\(2(84-7x)=-3x + 69\).
Expand the left - hand side: \(168-14x=-3x + 69\).
Add \(14x\) to both sides: \(168=-3x + 69+14x\).
Simplify the right - hand side: \(168 = 11x+69\).
Subtract 69 from both sides: \(168 - 69=11x\), so \(99 = 11x\).
Divide both sides by 11: \(x = 9\).

Step4: Find the length of \(NO\)

Substitute \(x = 9\) into the expression for \(NO\): \(NO=84-7x\).
\(NO=84-7\times9\).
\(NO=84 - 63\).

Answer:

\(21\)