ng triangle similarity etermining relationships within a triangle points s, u, and t are the midpoints of the sides of △pqr. which statements are correct? choose two correct answers. $\overline{su} \parallel \overline{rp}$ $su = pr$ $\frac{1}{2}ts = rq$ $\overline{ut} \
ot\parallel \overline{rp}$ $\frac{1}{2}qp = ut$

Question

Accepted Answer

# Explanation:
## Step1: Recall Midline Theorem
The Midline (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
- **Statement 1: $\overline{SU} \parallel \overline{RP}$**
  Since $ S $ and $ U $ are midpoints (given: $ S, U, T $ are midpoints of $ \triangle PQR $'s sides), by the Midline Theorem, $ SU $ connects midpoints of $ PQ $ and $ QR $, so it should be parallel to $ PR $ (or $ RP $). This statement is correct.

- **Statement 2: $ SU = PR $**
  By the Midline Theorem, $ SU = \frac{1}{2}PR $, so $ SU
eq PR $. This is incorrect.

- **Statement 3: $ \frac{1}{2}TS = RQ $**
  $ T $ and $ S $ are midpoints, so $ TS $ connects midpoints of $ PR $ and $ PQ $, so $ TS = \frac{1}{2}RQ $ (not $ \frac{1}{2}TS = RQ $). Rearranging, $ TS = \frac{1}{2}RQ \implies RQ = 2TS $, so this is incorrect.

- **Statement 4: $ \overline{UT}
ot\parallel \overline{RP} $**
  $ U $ and $ T $ are midpoints, so $ UT $ connects midpoints of $ QR $ and $ PR $, so $ UT \parallel PQ $, not $ RP $. Wait, no—wait, $ T $ is midpoint of $ PR $, $ U $ is midpoint of $ QR $, so $ UT \parallel PQ $. But the statement says $ UT
ot\parallel RP $. Wait, maybe I misread. Wait, $ RP $ is a side. Wait, actually, $ UT $ is parallel to $ PQ $, so it is not parallel to $ RP $? Wait, no—let's recheck. Wait, $ T $ (midpoint of $ PR $) and $ U $ (midpoint of $ QR $): by Midline Theorem, $ UT \parallel PQ $. So $ UT $ is not parallel to $ RP $ (since $ RP $ and $ PQ $ are sides of the triangle, meeting at $ P $, so they are not parallel). Wait, but the statement says $ \overline{UT}
ot\parallel \overline{RP} $, which is true? Wait, no, maybe I made a mistake. Wait, let's check the last statement.

- **Statement 5: $ \frac{1}{2}QP = UT $**
  $ U $ and $ T $ are midpoints: $ U $ (midpoint of $ QR $), $ T $ (midpoint of $ PR $)? Wait, no—wait, the triangle is $ \triangle PQR $, with vertices $ P, Q, R $. So sides: $ PQ, QR, PR $. $ S $ is midpoint of $ PQ $, $ U $ midpoint of $ QR $, $ T $ midpoint of $ PR $. Then:
  - $ SU $ connects $ S $ (mid $ PQ $) and $ U $ (mid $ QR $): parallel to $ PR $, length $ \frac{1}{2}PR $.
  - $ UT $ connects $ U $ (mid $ QR $) and $ T $ (mid $ PR $): parallel to $ PQ $, length $ \frac{1}{2}PQ $ (since $ PQ $ is the third side). So $ UT = \frac{1}{2}QP $ (since $ QP = PQ $), so $ \frac{1}{2}QP = UT $. This is correct.

Wait, earlier mistake: for Statement 4, $ UT \parallel PQ $, so $ UT $ is not parallel to $ RP $, so $ \overline{UT}
ot\parallel \overline{RP} $ is true? But the correct answers are two. Wait, let's re-express:

Wait, the first statement: $ \overline{SU} \parallel \overline{RP} $: $ S $ (mid $ PQ $), $ U $ (mid $ QR $): so $ SU \parallel PR $ (which is $ RP $), so that's correct.

The fifth statement: $ \frac{1}{2}QP = UT $: $ UT $ connects midpoints of $ QR $ and $ PR $, so $ UT \parallel PQ $ and $ UT = \frac{1}{2}PQ = \frac{1}{2}QP $, so $ \frac{1}{2}QP = UT $ is correct.

Wait, so the two correct statements are the first ($ \overline{SU} \parallel \overline{RP} $) and the fifth ($ \frac{1}{2}QP = UT $). Wait, but let's confirm:

- $ SU \parallel RP $: correct (Midline Theorem, $ SU $ midline of $ \triangle PQR $, parallel to $ PR $).
- $ \frac{1}{2}QP = UT $: $ UT $ is midline of $ \triangle PQR $ (connecting midpoints of $ QR $ and $ PR $), so $ UT \parallel PQ $ and $ UT = \frac{1}{2}PQ = \frac{1}{2}QP $, so correct.

# Answer:
The two correct statements are:
- $ \overline{SU} \parallel \overline{RP} $
- $ \frac{1}{2}QP = UT $

(Assuming the options are labeled as:
1. $ \overline{SU} \parallel \overline{RP} $
2. $ SU = PR $
3. $ \frac{1}{2}TS = RQ $
4. $ \overline{UT}
ot\parallel \overline{RP} $
5. $ \frac{1}{2}QP = UT $

Then the correct answers are the first and fifth options.)

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QUESTION IMAGE

Question

Explanation:

Step1: Recall Midline Theorem

Step2: Analyze Each Statement

Answer: