Question

solve for x.

Explanation:

Step1: Identify Midsegment Theorem

In a triangle, the midsegment (a segment connecting midpoints of two sides) is parallel to the third side and half its length. Here, \( U \) is midpoint of \( GH \) (since \( GU = UH \)) and \( T \) is midpoint of \( GF \) (since \( GT = TF \)), so \( UT \) is midsegment. Thus, \( UT=\frac{1}{2}FH \).

Step2: Set Up Equation

Given \( UT = x + 15 \) and \( FH = x + 21 \), substitute into midsegment formula:
\( x + 15=\frac{1}{2}(x + 21) \)

Step3: Solve for \( x \)

Multiply both sides by 2: \( 2(x + 15)=x + 21 \)
Expand left side: \( 2x + 30=x + 21 \)
Subtract \( x \) from both sides: \( 2x - x+ 30=x - x+ 21 \)
Simplify: \( x + 30=21 \)
Subtract 30 from both sides: \( x + 30 - 30=21 - 30 \)
Simplify: \( x=-9 \)

Answer:

\( x = -9 \)