Question

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p is the midpoint of \\(\overline{qt}\\) and r is the midpoint of \\(\overline{qs}\\). if \\(st = u + 39\\) and \\(pr = u\\), what is the value of \\(u\\)?

Explanation:

Step1: Identify Midsegment Theorem

Since \( P \) is the midpoint of \( \overline{QT} \) and \( R \) is the midpoint of \( \overline{QS} \), \( \overline{PR} \) is the midsegment of \( \triangle QST \). By the Midsegment Theorem, the midsegment is parallel to the third side and half its length. So, \( PR=\frac{1}{2}ST \).

Step2: Substitute Given Expressions

We know \( ST = u + 39 \) and \( PR = u \). Substitute these into the midsegment formula: \( u=\frac{1}{2}(u + 39) \).

Step3: Solve for \( u \)

Multiply both sides by 2 to eliminate the fraction: \( 2u=u + 39 \). Subtract \( u \) from both sides: \( 2u - u=u + 39 - u \), which simplifies to \( u = 39 \).

Answer:

\( 39 \)