Question

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v is the midpoint of \\(\overline{uw}\\) and t is the midpoint of \\(\overline{su}\\). if \\(sw = x + 39\\) and \\(tv = 2x\\), what is the value of x? image of a triangle with vertices u, s, w; t is on su, v is on uw, and segment tv is drawn. \\(x = \square\\)

Explanation:

Step1: Apply Midline Theorem

In triangle \( SUW \), \( T \) is the midpoint of \( SU \) and \( V \) is the midpoint of \( UW \). 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, \( TV \) is the midline, which means \( TV=\frac{1}{2}SW \).

Step2: Substitute the given expressions

We know \( SW = x + 39 \) and \( TV = 2x \). Substituting into the midline theorem equation: \( 2x=\frac{1}{2}(x + 39) \).

Step3: Solve for \( x \)

Multiply both sides by 2 to eliminate the fraction: \( 4x=x + 39 \).
Subtract \( x \) from both sides: \( 3x=39 \).
Divide both sides by 3: \( x = 13 \).

Answer:

\( 13 \)