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.
$\frac{1}{2} qp = ut$
$\frac{1}{2} ts = rq$
$su = pr$
$overline{su} || overline{rp}$
$overline{ut} perp overline{rp}$

Explanation:

Step1: Recall the mid - segment theorem

The mid - segment of a triangle is parallel to the third side and half its length.

Step2: Analyze each option

• For $\frac{1}{2}QP = UT$:

Since \(U\) and \(T\) are mid - points, \(UT\) is a mid - segment. By the mid - segment theorem, \(UT=\frac{1}{2}QP\).

• For $\frac{1}{2}TS = RQ$:

\(TS\) is not a mid - segment related to \(RQ\) in the correct proportion. In fact, if \(S\) and \(T\) were mid - points in a different configuration (not the case here for the relation with \(RQ\)), the mid - segment theorem does not support this.

• For \(SU = PR\):

\(SU\) is a mid - segment. By the mid - segment theorem, \(SU=\frac{1}{2}PR
eq PR\).

• For \(\overline{SU}\parallel\overline{RP}\):

Since \(S\) and \(U\) are mid - points, by the mid - segment theorem, \(SU\parallel RP\).

• For \(\overline{UT}\perp\overline{RP}\):

There is no information (such as right - angled triangle properties or given perpendicularity conditions) to suggest that \(UT\perp RP\). The mid - segment theorem only gives a parallel and length - proportion relationship, not a perpendicular one.

Answer:

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