Question

in $\triangle prs$, $qt$ connects the midpoints of $pr$ and $ps$. which of the following is true?
$qt \parallel rs$ and $qt = rs$
$qt \perp rs$ and $qt = \frac{1}{2}rs$
$qt \parallel rs$ and $qt = \frac{1}{2}rs$

Explanation:

Step1: Recall Midline Theorem

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

Step2: Apply to Given Triangle

In $\triangle PRS$, $Q$ is the midpoint of $PR$ and $T$ is the midpoint of $PS$. So, $QT$ is the midline of $\triangle PRS$ with respect to side $RS$. By the Midline Theorem, $QT \parallel RS$ and $QT = \frac{1}{2}RS$.

Step3: Evaluate Options

• First option: Claims $QT = RS$, which contradicts the theorem (should be half).
• Second option: Claims $QT \perp RS$, but the theorem says parallel, not perpendicular.
• Third option: Matches the Midline Theorem conclusion.

Answer:

The option "QT $\parallel$ RS and QT $= \frac{1}{2}$RS" (the third option among the given choices)