Question

points s, u, and t are the midpoints of the sides of △ pqr. which statements are correct? check all that apply. □ $\frac{1}{2}$qp = ut □ $\frac{1}{2}$ts = rq □ su = pr □ $overline{su} parallel overline{rp}$ □ $overline{ut} perp overline{rp}$

Explanation:

To solve this, we use the Midline Theorem (also known as the Midsegment Theorem) for triangles, which states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half as long.

Step 1: Analyze \(\boldsymbol{\frac{1}{2}QP = UT}\)

• \(U\) is the midpoint of \(RQ\) and \(T\) is the midpoint of \(RP\) (since \(S, U, T\) are midpoints). By the Midline Theorem, \(UT\) connects midpoints of \(RQ\) and \(RP\), so \(UT \parallel QP\) and \(UT=\frac{1}{2}QP\). So this statement is correct.

Step 2: Analyze \(\boldsymbol{\frac{1}{2}TS = RQ}\)

• \(T\) is the midpoint of \(RP\) and \(S\) is the midpoint of \(QP\). By the Midline Theorem, \(TS\) connects midpoints of \(RP\) and \(QP\), so \(TS \parallel RQ\) and \(TS = \frac{1}{2}RQ\), which implies \(RQ = 2TS\) or \(\frac{1}{2}RQ=TS\), not \(\frac{1}{2}TS = RQ\). So this statement is incorrect.

Step 3: Analyze \(\boldsymbol{SU = PR}\)

• \(S\) is the midpoint of \(QP\) and \(U\) is the midpoint of \(RQ\). By the Midline Theorem, \(SU\) connects midpoints of \(QP\) and \(RQ\), so \(SU \parallel PR\) and \(SU=\frac{1}{2}PR\), not \(SU = PR\). So this statement is incorrect.

Step 4: Analyze \(\boldsymbol{\overline{SU} \parallel \overline{RP}}\)

• From the Midline Theorem (as in Step 3), \(SU\) connects midpoints of \(QP\) and \(RQ\), so \(SU \parallel RP\) (since \(RP\) is the third side). So this statement is correct.

Step 5: Analyze \(\boldsymbol{\overline{UT} \perp \overline{RP}}\)

• From Step 1, \(UT \parallel QP\). There's no information given that \(QP \perp RP\), so we can't conclude \(UT \perp RP\). So this statement is incorrect.

Answer:

\(\boldsymbol{\frac{1}{2}QP = UT}\), \(\boldsymbol{\overline{SU} \parallel \overline{RP}}\) (i.e., the first and fourth options)