Question

in the diagram below of triangle ghi, j is the midpoint of \\(\overline{gi}\\) and k is the midpoint of \\(\overline{hi}\\). if \\(jk = 7x + 5\\), and \\(gh = 6x + 18\\), what is the measure of \\(\overline{jk}\\)?

Explanation:

Step1: Apply Midline Theorem

In triangle \( GHI \), \( J \) is the midpoint of \( \overline{GI} \) and \( K \) is the midpoint of \( \overline{HI} \). By the Midline Theorem, the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length. So, \( JK=\frac{1}{2}GH \).

Step2: Substitute the given expressions

We know \( JK = 7x + 5 \) and \( GH=6x + 18 \). Substitute into the midline theorem equation:
\( 7x + 5=\frac{1}{2}(6x + 18) \)

Step3: Solve for \( x \)

First, simplify the right - hand side: \( \frac{1}{2}(6x + 18)=3x + 9 \)
So the equation becomes \( 7x+5 = 3x + 9 \)
Subtract \( 3x \) from both sides: \( 7x-3x+5=3x - 3x+9 \), which gives \( 4x+5 = 9 \)
Subtract 5 from both sides: \( 4x+5 - 5=9 - 5 \), so \( 4x=4 \)
Divide both sides by 4: \( x = 1 \)

Step4: Find the length of \( JK \)

Substitute \( x = 1 \) into the expression for \( JK \): \( JK=7(1)+5=7 + 5=12 \)

Answer:

The measure of \( \overline{JK} \) is \( 12 \).