Question

determining relationships within a triangle
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:

Step1: Recall Midline Theorem

The Midline Theorem (or Midsegment Theorem) states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half as long.

Step2: Analyze Each Statement

• For $\frac{1}{2}QP = UT$: Since $U$ and $T$ are midpoints, $UT$ is the midline parallel to $QP$? Wait, no, wait. Wait, $S, U, T$ are midpoints. Let's label the triangle: $PQR$, with midpoints $S$ (midpoint of $PQ$), $U$ (midpoint of $RQ$), $T$ (midpoint of $PR$)? Wait, no, looking at the diagram: $T$ is on $PR$, $S$ is on $PQ$, $U$ is on $RQ$. So by Midline Theorem:
• $UT$: connects midpoints of $RQ$ and $PR$? Wait, no. Wait, $S$ is midpoint of $PQ$, $U$ midpoint of $RQ$, $T$ midpoint of $PR$. Then:
• $UT$: midline of $\triangle PRQ$? Wait, $U$ (mid $RQ$), $T$ (mid $PR$), so $UT$ is parallel to $PQ$ and $UT = \frac{1}{2}PQ$. So $\frac{1}{2}QP = UT$: correct.
• $\frac{1}{2}TS = RQ$: $TS$: $T$ (mid $PR$), $S$ (mid $PQ$), so $TS$ is midline parallel to $RQ$ and $TS = \frac{1}{2}RQ$, so $\frac{1}{2}TS = \frac{1}{4}RQ

eq RQ$. So incorrect.

• $SU = PR$: $SU$: $S$ (mid $PQ$), $U$ (mid $RQ$), so $SU$ is midline parallel to $PR$ and $SU = \frac{1}{2}PR

eq PR$. So incorrect.

• $\overline{SU} \parallel \overline{RP}$: $SU$ connects mid $PQ$ and mid $RQ$, so by Midline Theorem, $SU \parallel PR$ (since $PR$ is the third side). So correct.
• $\overline{UT} \perp \overline{RP}$: There's no indication of right angle. $UT$ is parallel to $PQ$, not necessarily perpendicular to $RP$. So incorrect.

Step3: Identify Correct Statements

From above, $\frac{1}{2}QP = UT$ and $\overline{SU} \parallel \overline{RP}$ are correct. Wait, let's recheck:

• $UT$: midline between $RQ$? No, wait, $U$ is mid $RQ$, $T$ is mid $PR$, so $UT$ is midline of $\triangle PRQ$, so $UT \parallel PQ$ and $UT = \frac{1}{2}PQ$. So $\frac{1}{2}QP = UT$: correct.
• $SU$: $S$ (mid $PQ$), $U$ (mid $RQ$), so $SU$ is midline of $\triangle PQR$, connecting mid $PQ$ and mid $RQ$, so $SU \parallel PR$ (since $PR$ is the side opposite to the midline). So $\overline{SU} \parallel \overline{RP}$: correct.

Wait, also, let's check again:

• $\frac{1}{2}QP = UT$: yes, because $UT$ is midline, so length half of $PQ$.
• $\overline{SU} \parallel \overline{RP}$: yes, by Midline Theorem, $SU$ parallel to $PR$.

Wait, what about the other options:

• $SU = PR$: $SU$ is half of $PR$, so no.
• $\frac{1}{2}TS = RQ$: $TS$ is half of $RQ$, so $\frac{1}{2}TS = \frac{1}{4}RQ$, no.
• $UT \perp RP$: no, no right angle given.

So correct statements: $\frac{1}{2}QP = UT$ and $\overline{SU} \parallel \overline{RP}$. Wait, but let's check the options again:
Options:

1. $\frac{1}{2}QP = UT$: correct.
2. $\frac{1}{2}TS = RQ$: incorrect.
3. $SU = PR$: incorrect.
4. $\overline{SU} \parallel \overline{RP}$: correct.
5. $\overline{UT} \perp \overline{RP}$: incorrect.

So the correct ones are the first and fourth options.

Answer:

$\frac{1}{2}QP = UT$, $\overline{SU} \parallel \overline{RP}$ (the first and fourth checkboxes: $\boldsymbol{\frac{1}{2}QP = UT}$ and $\boldsymbol{\overline{SU} \parallel \overline{RP}}$)