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}paralleloverline{rp})

(overline{ut}perpoverline{rp})

Explanation:

Step1: Use the mid - segment theorem

The mid - segment theorem of a triangle states that the segment connecting the mid - points of two sides of a triangle is parallel to the third side and half its length.
In $\triangle PQR$, since $U$ and $T$ are mid - points of two sides, $UT$ is a mid - segment. So, $UT\parallel QP$ and $UT=\frac{1}{2}QP$.
In $\triangle PQR$, since $S$ and $U$ are mid - points of two sides, $SU$ is a mid - segment. So, $SU\parallel RP$ and $SU = \frac{1}{2}RP$.

Step2: Analyze each option

• For $\frac{1}{2}QP = UT$: By the mid - segment theorem (as $U$ and $T$ are mid - points), this is correct.
• For $\frac{1}{2}TS=RQ$: There is no theorem to support this. $TS$ is not related to $RQ$ in this proportion by the mid - segment theorem.
• For $SU = PR$: By the mid - segment theorem, $SU=\frac{1}{2}PR$, not $SU = PR$.
• For $\overline{SU}\parallel\overline{RP}$: By the mid - segment theorem (as $S$ and $U$ are mid - points), this is correct.
• For $\overline{UT}\perp\overline{RP}$: There is no information (such as right - angled triangle conditions) to suggest that $UT$ is perpendicular to $RP$. The mid - segment theorem only gives a parallel and length - ratio relationship, not a perpendicular relationship.

Answer:

$\frac{1}{2}QP = UT$, $\overline{SU}\parallel\overline{RP}$