in the figure below, points s, t, and u are the midpoints of the sides of △pqr. suppose pr = 70, pq = 74, and su = 12. find the following lengths. qr = qt = st =

Question

in the figure below, points s, t, and u are the midpoints of the sides of △pqr. suppose pr = 70, pq = 74, and su = 12. find the following lengths. qr =  qt =  st =

Accepted Answer

# 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: Find $ QR $
Points $ S $ and $ U $ are midpoints. So $ SU $ is the midline parallel to $ QR $, so $ SU=\frac{1}{2}QR $. Given $ SU = 12 $, then $ QR=2\times SU=2\times12 = 24 $? Wait, no, wait. Wait, maybe I mixed up. Wait, let's check the triangle. Wait, $ S $ is midpoint of $ PQ $, $ U $ is midpoint of $ PR $, $ T $ is midpoint of $ QR $. Wait, maybe. Wait, let's re - identify the midpoints.

Wait, the problem says $ S $, $ T $, $ U $ are midpoints of the sides of $ \triangle PQR $. Let's assume: $ S $ is midpoint of $ PQ $, $ U $ is midpoint of $ PR $, $ T $ is midpoint of $ QR $.

Then, the midline between $ S $ (midpoint of $ PQ $) and $ U $ (midpoint of $ PR $): $ SU \parallel QR $ and $ SU=\frac{1}{2}QR $. Wait, but we have $ SU = 12 $, so $ QR=2\times SU = 24 $? Wait, no, that can't be. Wait, maybe $ S $ is midpoint of $ PQ $, $ T $ is midpoint of $ QR $, $ U $ is midpoint of $ PR $.

Wait, let's find $ QT $: $ T $ is midpoint of $ QR $, so $ QT=\frac{1}{2}QR $? No, wait, $ T $ is midpoint of $ QR $, so $ QT=\frac{1}{2}QR $ only if $ Q $ to $ T $ is half of $ QR $. Wait, no, $ T $ is midpoint, so $ QT = TR=\frac{1}{2}QR $.

Wait, let's find $ ST $: $ S $ is midpoint of $ PQ $, $ T $ is midpoint of $ QR $, so $ ST $ is midline parallel to $ PR $ and $ ST=\frac{1}{2}PR $. Given $ PR = 70 $, so $ ST=\frac{1}{2}\times70=35 $.

Wait, let's correct the earlier mistake. Let's re - assign the midpoints properly.

Let's assume:

- $ S $ is the midpoint of $ PQ $
- $ U $ is the midpoint of $ PR $
- $ T $ is the midpoint of $ QR $

Then:

1. For $ ST $: $ S $ (midpoint of $ PQ $) and $ T $ (midpoint of $ QR $): By Midline Theorem, $ ST\parallel PR $ and $ ST = \frac{1}{2}PR $. Given $ PR = 70 $, so $ ST=\frac{70}{2}=35 $.

2. For $ QT $: $ T $ is the midpoint of $ QR $, but we need another relation. Wait, $ S $ is midpoint of $ PQ $, $ U $ is midpoint of $ PR $, so $ SU\parallel QR $ and $ SU=\frac{1}{2}QR $. Given $ SU = 12 $, so $ QR = 2\times SU=24 $? No, that seems small. Wait, maybe $ S $ is midpoint of $ PQ $, $ T $ is midpoint of $ QR $, $ U $ is midpoint of $ PR $.

Wait, $ PQ = 74 $, $ S $ is midpoint of $ PQ $, so $ PS=SQ=\frac{PQ}{2}=\frac{74}{2}=37 $.

Wait, let's start over.

We know:

- $ S $, $ T $, $ U $ are midpoints.

Case 1: Find $ QT $

Since $ T $ is the midpoint of $ QR $? No, wait, maybe $ T $ is the midpoint of $ PR $? No, the problem says "midpoints of the sides". The sides are $ PQ $, $ QR $, $ PR $.

So three sides: $ PQ $, $ QR $, $ PR $. So three midpoints: one for each side.

Let's suppose:

- $ S $: midpoint of $ PQ $
- $ T $: midpoint of $ QR $
- $ U $: midpoint of $ PR $

Then:

- $ ST $: connects midpoints of $ PQ $ and $ QR $, so $ ST\parallel PR $ and $ ST=\frac{1}{2}PR $. Given $ PR = 70 $, so $ ST=\frac{70}{2}=35 $.

- $ QT $: $ T $ is midpoint of $ QR $, so $ QT=\frac{1}{2}QR $. But we need to find $ QR $.

- $ SU $: connects midpoints of $ PQ $ and $ PR $, so $ SU\parallel QR $ and $ SU=\frac{1}{2}QR $. Given $ SU = 12 $, so $ QR = 2\times SU=2\times12 = 24 $? Wait, that seems odd, but let's check.

Wait, no, if $ S $ is midpoint of $ PQ $, $ U $ is midpoint of $ PR $, then $ SU $ is midline to $ QR $, so $ SU=\frac{1}{2}QR $, so $ QR = 2SU=24 $. Then $ QT=\frac{1}{2}QR=\frac{24}{2}=12 $? No, that can't be. Wait, maybe my assumption of the midpoints is wrong.

Wait, let's look at the figure (even though we can't see it, but from the labels: $ P $, $ Q $, $ R $ form a triangle. $ S $ is on $ PQ $, $ U $ is on $ PR $, $ T $ is on $ QR $.

Let's find $ QT $: $ T $ is midpoint of $ QR $? No, maybe $ T $ is midpoint of $ PR $? No, the problem says midpoints of the sides.

Wait, $ PQ = 74 $, $ S $ is midpoint of $ PQ $, so $ SQ=\frac{PQ}{2}=37 $.

Wait, let's use the midline theorem correctly.

1. Find $ ST $:

If $ S $ is midpoint of $ PQ $ and $ T $ is midpoint of $ QR $, then $ ST \parallel PR $ and $ ST=\frac{1}{2}PR $. Since $ PR = 70 $, $ ST=\frac{70}{2}=35 $.

2. Find $ QT $:

If $ T $ is midpoint of $ PR $? No, $ PR = 70 $, so midpoint would be 35, but $ QT $ is a segment from $ Q $ to $ T $. Wait, maybe $ T $ is midpoint of $ QR $, no. Wait, maybe $ S $ is midpoint of $ PQ $, $ U $ is midpoint of $ PR $, $ T $ is midpoint of $ QR $.

Then:

- $ SU \parallel QR $, $ SU=\frac{1}{2}QR $, so $ QR = 2SU=24 $

- $ ST \parallel PR $, $ ST=\frac{1}{2}PR = 35 $

- $ QT $: since $ T $ is midpoint of $ QR $, $ QT=\frac{1}{2}QR = 12 $? No, that doesn't make sense. Wait, maybe I mixed up the midpoints.

Wait, let's try another approach.

We know $ PQ = 74 $, $ S $ is midpoint of $ PQ $, so $ PS=SQ=\frac{74}{2}=37 $.

$ PR = 70 $, $ U $ is midpoint of $ PR $, so $ PU=UR=\frac{70}{2}=35 $.

Now, $ SU $ is a midline (connecting midpoints of $ PQ $ and $ PR $), so $ SU\parallel QR $ and $ SU=\frac{1}{2}QR $. Given $ SU = 12 $, so $ QR = 2\times12 = 24 $.

$ T $ is midpoint of $ QR $, so $ QT=\frac{1}{2}QR=\frac{24}{2}=12 $? No, that can't be. Wait, maybe $ T $ is midpoint of $ PR $? No, $ PR = 70 $, midpoint is 35.

Wait, $ ST $: if $ S $ is midpoint of $ PQ $ and $ T $ is midpoint of $ PR $, then $ ST\parallel QR $ and $ ST=\frac{1}{2}QR $. But $ SU $ is midpoint of $ PQ $ and $ U $ is midpoint of $ PR $, so $ SU\parallel QR $, same as $ ST $. That can't be.

Wait, maybe the figure is such that $ S $ is midpoint of $ PQ $, $ T $ is midpoint of $ QR $, $ U $ is midpoint of $ PR $.

Then:

- $ ST $: midline of $ \triangle PQR $ (connecting midpoints of $ PQ $ and $ QR $), so $ ST\parallel PR $ and $ ST=\frac{1}{2}PR=\frac{70}{2}=35 $

- $ QT $: $ T $ is midpoint of $ QR $, so $ QT=\frac{1}{2}QR $. But we need to find $ QR $. Since $ SU $ is midline (connecting midpoints of $ PQ $ and $ PR $), $ SU\parallel QR $ and $ SU=\frac{1}{2}QR $, so $ QR = 2SU = 24 $, so $ QT=\frac{24}{2}=12 $

- $ QR = 24 $

Wait, but let's verify:

If $ QR = 24 $, $ PQ = 74 $, $ PR = 70 $, then the triangle has sides 74, 70, 24. Let's check the midlines:

- Midline between $ PQ $ and $ PR $ ( $ SU $): $ \frac{1}{2}QR=\frac{24}{2}=12 $, which matches $ SU = 12 $.

- Midline between $ PQ $ and $ QR $ ( $ ST $): $ \frac{1}{2}PR=\frac{70}{2}=35 $

- Midline between $ PR $ and $ QR $ ( $ UT $): $ \frac{1}{2}PQ=\frac{74}{2}=37 $

And $ T $ is midpoint of $ QR $, so $ QT=\frac{1}{2}QR = 12 $? Wait, no, $ QR = 24 $, so $ QT = 12 $, $ ST = 35 $, $ QR = 24 $

So:

- $ QR=24 $

- $ QT = 12 $ (since $ T $ is midpoint of $ QR $, $ QT=\frac{1}{2}QR $)

- $ ST = 35 $ (since $ ST $ is midline parallel to $ PR $, $ ST=\frac{1}{2}PR $)

Wait, but let's check the midpoint of $ PQ $: $ S $ is midpoint, so $ SQ=\frac{74}{2}=37 $, but $ QT = 12 $, that's okay.

So:

1. $ QR $: Since $ SU $ is midline ( $ S $ midpoint of $ PQ $, $ U $ midpoint of $ PR $ ), $ SU=\frac{1}{2}QR $, so $ QR = 2\times SU=2\times12 = 24 $

2. $ QT $: $ T $ is midpoint of $ QR $, so $ QT=\frac{1}{2}QR=\frac{24}{2}=12 $

3. $ ST $: $ S $ midpoint of $ PQ $, $ T $ midpoint of $ QR $, so $ ST $ is midline parallel to $ PR $, $ ST=\frac{1}{2}PR=\frac{70}{2}=35 $

# Answer:
$ QR = 24 $, $ QT = 12 $, $ ST = 35 $

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QUESTION IMAGE

Question

Explanation:

Step1: Recall Midline Theorem

Step2: Find \( QR \)

Answer: