m, n, and t are the midpoints of the sides of △xyz.
2. if xz = 10, then mn = _?_.
3. if tn = 7, then xy = _?_.
4. if zn = 8, then tm = _?_.
5. if xy = k, then tn = _?_.
6. suppose xy = 10, yz = 14, and xz = 8.
what are the lengths of the three sides of
a. △tnz? b. △myn?
c. △xmt? d. △ntm?
7. state a theorem suggested by exercise 6.
8. how many parallelograms are in the diagram?
(there is a diagram on the right: △xyz with t, n, m as midpoints, forming some segments inside.)

Question

m, n, and t are the midpoints of the sides of △xyz.
2. if xz = 10, then mn = _?_.
3. if tn = 7, then xy = _?_.
4. if zn = 8, then tm = _?_.
5. if xy = k, then tn = _?_.
6. suppose xy = 10, yz = 14, and xz = 8.
   what are the lengths of the three sides of
   a. △tnz?  b. △myn?
   c. △xmt?  d. △ntm?
7. state a theorem suggested by exercise 6.
8. how many parallelograms are in the diagram?
(there is a diagram on the right: △xyz with t, n, m as midpoints, forming some segments inside.)

Accepted Answer

# Explanation:
## Step1: Recall Midsegment Theorem
The Midsegment Theorem (or Midline 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: Solve Question 2
Given $ M, N, T $ are midpoints of $ \triangle XYZ $. For $ MN $: $ M $ is midpoint of $ XY $, $ N $ is midpoint of $ YZ $? Wait, no, let's check the diagram. Wait, $ T $ is midpoint of $ XZ $? Wait, no, the diagram: $ T $ on $ XZ $, $ M $ on $ XY $, $ N $ on $ YZ $? Wait, no, the problem says $ M, N, T $ are midpoints of the sides. So by Midsegment Theorem, $ MN $ connects midpoints of $ XY $ and $ YZ $? Wait, no, $ XZ = 10 $, we need $ MN $. Wait, actually, $ M $ is midpoint of $ XY $, $ N $ is midpoint of $ YZ $, $ T $ is midpoint of $ XZ $. Wait, no, let's re-express: The midsegment between $ XZ $ and $ XY $? Wait, no, $ MN $: if $ M $ is midpoint of $ XY $, $ N $ is midpoint of $ YZ $, then $ MN $ is midsegment parallel to $ XZ $ and $ MN=\frac{1}{2}XZ $. Yes, that makes sense. So $ XZ = 10 $, so $ MN=\frac{1}{2}\times10 = 5 $.

## Step3: Solve Question 3
$ TN = 7 $. $ T $ is midpoint of $ XZ $, $ N $ is midpoint of $ YZ $? Wait, no, $ TN $: $ T $ midpoint of $ XZ $, $ N $ midpoint of $ YZ $? Wait, no, $ XY $ is the third side. Wait, $ TN $ connects midpoints of $ XZ $ and $ YZ $, so it should be parallel to $ XY $ and $ TN=\frac{1}{2}XY $. So if $ TN = 7 $, then $ XY = 2\times7 = 14 $.

## Step4: Solve Question 4
$ ZN = 8 $. $ ZN $ is half of $ YZ $ (since $ N $ is midpoint), so $ YZ = 16 $. $ TM $: $ T $ midpoint of $ XZ $, $ M $ midpoint of $ XY $, so $ TM $ is midsegment parallel to $ YZ $, so $ TM=\frac{1}{2}YZ $. Thus $ TM=\frac{1}{2}\times16 = 8 $.

## Step5: Solve Question 5
$ XY = k $. $ TN $ is midsegment parallel to $ XY $, so $ TN=\frac{1}{2}XY=\frac{k}{2} $.

## Step6: Solve Question 6a: $ \triangle TNZ $
$ XY = 10 $, $ YZ = 14 $, $ XZ = 8 $. $ T $ is midpoint of $ XZ $, so $ XT = TZ=\frac{8}{2}=4 $. $ N $ is midpoint of $ YZ $, so $ ZN = NY=\frac{14}{2}=7 $. $ TN $ is midsegment, so $ TN=\frac{1}{2}XY=\frac{10}{2}=5 $. So sides of $ \triangle TNZ $: $ TZ = 4 $, $ ZN = 7 $, $ TN = 5 $.

## Step6b: $ \triangle MYN $
$ M $ is midpoint of $ XY $, so $ XM = MY=\frac{10}{2}=5 $. $ N $ is midpoint of $ YZ $, $ ZN = NY = 7 $. $ MN $ is midsegment, so $ MN=\frac{1}{2}XZ=\frac{8}{2}=4 $. So sides: $ MY = 5 $, $ YN = 7 $, $ MN = 4 $.

## Step6c: $ \triangle XMT $
$ XM = 5 $ (midpoint of $ XY $), $ XT = 4 $ (midpoint of $ XZ $), $ TM $ is midsegment, $ TM=\frac{1}{2}YZ=\frac{14}{2}=7 $. So sides: $ XM = 5 $, $ XT = 4 $, $ TM = 7 $.

## Step6d: $ \triangle NTM $
$ TN = 5 $, $ TM = 7 $, $ MN = 4 $. Wait, let's check: $ TN = 5 $, $ TM = 7 $, $ MN = 4 $. So sides: $ TN = 5 $, $ TM = 7 $, $ MN = 4 $.

## Step7: Solve Question 7
The theorem suggested is the Midsegment Theorem: The segment connecting the midpoints of two sides of a triangle is parallel to the third side and half as long. Also, the four triangles formed by the midsegments are congruent? Wait, no, from Exercise 6, the three midsegments divide the original triangle into four smaller triangles, each similar to the original triangle and to each other, with side lengths half of the original. Wait, but the main theorem here is the Midsegment Theorem, and also that the triangle formed by midsegments has sides half of the original, and the four triangles are congruent. But the key theorem is the Midsegment Theorem and the properties of midsegments creating similar triangles with half the side lengths.

## Step8: Solve Question 8
To find the number of parallelograms: In the diagram, the midsegments create parallelograms. Let's list them: $ XMTN $: $ XM \parallel TN $, $ XT \parallel MN $? Wait, no. Wait, $ TM \parallel YN $ (since $ TM $ is midsegment), $ TN \parallel MY $, $ MN \parallel XT $. Wait, actually, the parallelograms are: $ \square XMTN $, $ \square TMNY $, $ \square XMNT $? Wait, no, let's see: The midpoints form three midsegments, creating three parallelograms? Wait, no, the correct number is 3? Wait, no, actually, when you connect midpoints of a triangle, you form four smaller triangles and three parallelograms. Let's check: $ T $ (mid $ XZ $), $ M $ (mid $ XY $), $ N $ (mid $ YZ $). So $ TM \parallel YZ $, $ TN \parallel XY $, $ MN \parallel XZ $. So the parallelograms are: $ \square XMTN $ ( $ XM \parallel TN $, $ XT \parallel MN $ ), $ \square TMNY $ ( $ TM \parallel YN $, $ TN \parallel MY $ ), $ \square XMNT $? Wait, no, actually three parallelograms: $ XMTN $, $ TMNY $, and $ XMNT $? Wait, no, the standard result is that connecting midpoints of a triangle creates three parallelograms. Wait, let's count: $ T-M-N-T $: no, $ T $ to $ M $ to $ N $ to $ T $: is that a parallelogram? $ TM \parallel YZ $, $ TN \parallel XY $, $ MN \parallel XZ $. So the parallelograms are: 1. $ XMTN $ (sides $ XM, MT, TN, NX $? No, $ XM \parallel TN $, $ XT \parallel MN $), 2. $ TMNY $ ( $ TM \parallel YN $, $ TN \parallel MY $ ), 3. $ XMNT $? Wait, maybe I'm overcomplicating. The correct number is 3? Wait, no, actually, when you have three midsegments, the number of parallelograms is 3. Wait, let's see the diagram: $ T $ on $ XZ $, $ M $ on $ XY $, $ N $ on $ YZ $. So the parallelograms are: $ \square XMTN $, $ \square TMNY $, and $ \square XMNT $? No, actually, the three parallelograms are formed by each pair of midsegments. So the answer is 3? Wait, no, let's think again. The midsegment theorem: each midsegment is parallel to a side. So $ TM \parallel YZ $, $ TN \parallel XY $, $ MN \parallel XZ $. So the quadrilaterals: $ XMTN $: $ XM \parallel TN $ (since $ TN \parallel XY $ and $ XM $ is part of $ XY $), $ XT \parallel MN $ (since $ MN \parallel XZ $ and $ XT $ is part of $ XZ $): so that's a parallelogram. $ TMNY $: $ TM \parallel YN $ ( $ YN $ is part of $ YZ $, $ TM \parallel YZ $ ), $ TN \parallel MY $ ( $ MY $ is part of $ XY $, $ TN \parallel XY $ ): parallelogram. $ XMNT $? No, $ XM \parallel TN $, $ MN \parallel XT $: that's the same as $ XMTN $. Wait, maybe the correct number is 3? Wait, no, actually, the number of parallelograms in the midline triangle diagram is 3. Wait, let's check standard problems: when you connect midpoints of a triangle, you get three parallelograms. So the answer is 3.

# Answers (for each question):
2. $ \boldsymbol{5} $
3. $ \boldsymbol{14} $
4. $ \boldsymbol{8} $
5. $ \boldsymbol{\frac{k}{2}} $
6a. Sides of $ \triangle TNZ $: $ 4, 5, 7 $ ( $ TZ = 4 $, $ TN = 5 $, $ ZN = 7 $ )
6b. Sides of $ \triangle MYN $: $ 5, 4, 7 $ ( $ MY = 5 $, $ MN = 4 $, $ YN = 7 $ )
6c. Sides of $ \triangle XMT $: $ 5, 4, 7 $ ( $ XM = 5 $, $ XT = 4 $, $ TM = 7 $ )
6d. Sides of $ \triangle NTM $: $ 5, 7, 4 $ ( $ TN = 5 $, $ TM = 7 $, $ MN = 4 $ )
7. The Midsegment Theorem (or Midline Theorem) and the theorem that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length, and the three midsegments divide the original triangle into four smaller congruent triangles (or four triangles similar to the original with scale factor $ \frac{1}{2} $) and three parallelograms.
8. $ \boldsymbol{3} $

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Recall Midsegment Theorem

Step2: Solve Question 2

Step3: Solve Question 3

Step4: Solve Question 4

Step5: Solve Question 5

Step6: Solve Question 6a: \( \triangle TNZ \)

Step6b: \( \triangle MYN \)

Step6c: \( \triangle XMT \)

Step6d: \( \triangle NTM \)

Step7: Solve Question 7

Answer: