8. according to the midsegment theorem, what is the length of the midsegments in the diagram? diagram with triangle, segments labeled 24, 8, 10.6, points d, e, f, g, h, j options: jh = 8, gh = 12 and gj = 10.6; jh = 2, gh = 6 and gj = 2.8; jh = 16, gh = 48 and gj = 21.2; jh = 6, gh = 8 and gj = 3.2

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Accepted Answer

# Explanation:
## Step1: Recall Midsegment Theorem
The Midsegment Theorem (also known as the 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. In a triangle, if $ G $, $ H $, $ J $ are midpoints, then:
- $ JH $ is midsegment to $ DF $, so $ JH=\frac{1}{2}DF $. Given $ DF = 8 $, wait no, wait $ DF = 8 $? Wait, no, looking at the diagram: $ D F $ has length? Wait, no, the side $ DE $ is 24, $ DF = 8 $, $ E F=10.6 $. Wait, actually, $ G $, $ H $, $ J $ are midpoints. So:
- For $ JH $: $ JH $ is midsegment to $ DF $, so $ JH=\frac{1}{2}DF $. Wait $ DF = 8 $? No, wait $ DF $ is 8? Wait, no, the side $ DF $ is labeled 8, so $ JH=\frac{1}{2}\times8 = 4 $? Wait, no, maybe I misread. Wait, the top side $ DE $ is 24. So $ GH $ is midsegment to $ DE $, so $ GH=\frac{1}{2}DE $. $ DE = 24 $, so $ GH=\frac{1}{2}\times24 = 12 $. Then $ JH $: $ JH $ is midsegment to $ DF $, $ DF = 8 $, so $ JH=\frac{1}{2}\times8 = 4 $? No, wait the options: first option is $ JH = 8 $, $ GH = 12 $, $ GJ = 10.6 $. Wait, $ GJ $ is midsegment to $ EF $, $ EF = 10.6 $, so $ GJ=\frac{1}{2}\times10.6 = 5.3 $? No, that's not matching. Wait, maybe I got the midsegments wrong. Wait, the Midsegment Theorem: in a triangle, the midsegment is parallel to the third side and half its length. So if $ G $, $ H $, $ J $ are midpoints, then:
- $ GH $ is midsegment to $ DE $, so $ GH=\frac{1}{2}DE $. $ DE = 24 $, so $ GH = 12 $.
- $ JH $ is midsegment to $ DF $, $ DF = 8 $, so $ JH=\frac{1}{2}\times8 = 4 $? No, but the first option has $ JH = 8 $, $ GH = 12 $, $ GJ = 10.6 $. Wait, maybe $ DF $ is 16? Wait, no, the diagram: $ D $ to $ F $ is 8, $ F $ to $ E $? No, $ E $ to $ F $ is 10.6. Wait, maybe $ JH $ is midsegment to $ DF $, but $ DF $ is 8? No, that can't be. Wait, maybe I mixed up the sides. Wait, the triangle is $ DEF $, with $ D $, $ E $, $ F $. $ J $ is midpoint of $ DE $, $ G $ midpoint of $ DF $, $ H $ midpoint of $ EF $. Then:
- $ GJ $ is midsegment to $ EF $, so $ GJ=\frac{1}{2}EF $. $ EF = 10.6 $, so $ GJ=\frac{1}{2}\times10.6 = 5.3 $? No, the first option has $ GJ = 10.6 $. Wait, maybe the midsegments are equal to the sides they are midsegments of? No, midsegment is half. Wait, no, maybe the diagram is a triangle divided into four smaller triangles, so $ G $, $ H $, $ J $ are midpoints, so each midsegment is half the length of the corresponding side. Wait, let's check the first option: $ JH = 8 $, $ GH = 12 $, $ GJ = 10.6 $. Let's see:
- $ GH $: if $ GH = 12 $, and $ DE = 24 $, then $ GH=\frac{1}{2}DE $ (24/2=12), that works.
- $ JH $: if $ JH = 8 $, and $ DF = 16 $? Wait, no, $ DF $ is labeled 8. Wait, maybe $ DF = 16 $? No, the diagram shows $ DF = 8 $. Wait, maybe I misread $ DF $. Wait, the side $ DF $ is 8, but the first option has $ JH = 8 $, which would mean $ JH = DF $, but midsegment should be half. Wait, no, maybe the triangle is a different configuration. Wait, maybe $ JH $ is equal to $ DF $? No, midsegment is half. Wait, maybe the problem is that $ G $, $ H $, $ J $ are midpoints, so $ JH \parallel DF $ and $ JH = DF $? No, that's not the theorem. Wait, maybe the diagram is a triangle with $ DE = 24 $, $ DF = 8 $, $ EF = 10.6 $, and $ G $, $ H $, $ J $ are midpoints, so:
- $ GH $: midsegment to $ DE $, so $ GH = \frac{1}{2}DE = 12 $ (matches first option's $ GH = 12 $).
- $ JH $: midsegment to $ DF $, so $ JH = \frac{1}{2}DF $. If $ DF = 16 $, then $ JH = 8 $ (matches first option's $ JH = 8 $). Oh! Wait, maybe $ DF = 16 $, not 8. Maybe I misread the diagram. The label 8 is on $ DG $, not $ DF $. So $ DG = 8 $, so $ DF = 2\times DG = 16 $, because $ G $ is midpoint. Ah! That's the mistake. So $ G $ is midpoint of $ DF $, so $ DG = GF = 8 $, so $ DF = 16 $. Then $ JH $ is midsegment to $ DF $, so $ JH = \frac{1}{2}DF = \frac{1}{2}\times16 = 8 $ (matches first option's $ JH = 8 $). Then $ GJ $ is midsegment to $ EF $, $ EF = 10.6 $, so $ GJ = \frac{1}{2}EF = 5.3 $? No, first option has $ GJ = 10.6 $. Wait, no, $ H $ is midpoint of $ EF $, so $ EH = HF = \frac{1}{2}EF = 5.3 $, but $ GJ $ is midsegment to $ EF $, so $ GJ = EH = 10.6 $? Wait, no, $ EF = 10.6 $, so midsegment $ GJ = \frac{1}{2}EF = 5.3 $, but first option has $ GJ = 10.6 $. Wait, maybe $ EF = 21.2 $, but no, diagram shows $ EF = 10.6 $. Wait, maybe $ GJ $ is equal to $ EF $? No, midsegment is half. Wait, the first option: $ JH = 8 $, $ GH = 12 $, $ GJ = 10.6 $. Let's check:
- $ GH = 12 $: $ GH $ is midsegment to $ DE = 24 $, so $ GH = 12 $ (correct).
- $ JH = 8 $: $ JH $ is midsegment to $ DF $, if $ DF = 16 $, then $ JH = 8 $ (correct, since $ G $ is midpoint, $ DF = 2\times DG = 16 $).
- $ GJ = 10.6 $: $ GJ $ is midsegment to $ EF $, if $ EF = 21.2 $, but diagram shows $ EF = 10.6 $. Wait, no, $ H $ is midpoint of $ EF $, so $ EF = 2\times EH = 21.2 $? No, diagram shows $ EF = 10.6 $. Wait, maybe the diagram has $ EF = 10.6 $, so $ GJ = \frac{1}{2}EF = 5.3 $, but that's not in options. Wait, the first option has $ GJ = 10.6 $, which would mean $ GJ = EF $, so maybe $ GJ $ is not a midsegment but equal to $ EF $? No, midsegment should be half. Wait, maybe I got the midsegments reversed. Let's re-express:

In triangle $ DEF $, with midpoints $ G $ (of $ DF $), $ H $ (of $ EF $), $ J $ (of $ DE $):

- Midsegment $ GH \parallel DE $ and $ GH = \frac{1}{2}DE $. $ DE = 24 $, so $ GH = 12 $ (matches first option's $ GH = 12 $).
- Midsegment $ JH \parallel DF $ and $ JH = \frac{1}{2}DF $. $ DF = 16 $ (since $ DG = 8 $, $ G $ is midpoint, so $ DF = 16 $), so $ JH = 8 $ (matches first option's $ JH = 8 $).
- Midsegment $ GJ \parallel EF $ and $ GJ = \frac{1}{2}EF $. $ EF = 21.2 $? No, diagram shows $ EF = 10.6 $. Wait, no, $ EF = 10.6 $, so $ GJ = \frac{1}{2}\times10.6 = 5.3 $, but first option has $ GJ = 10.6 $. Wait, maybe $ EF = 10.6 $, and $ GJ = EF $, which would mean $ GJ $ is not a midsegment. But the problem says "midsegments", so all three should be midsegments. Wait, the first option is $ JH = 8 $, $ GH = 12 $, $ GJ = 10.6 $. Let's check the lengths:

- $ GH = 12 $: $ \frac{1}{2}\times24 = 12 $ (correct, $ DE = 24 $).
- $ JH = 8 $: $ \frac{1}{2}\times16 = 8 $ (correct, $ DF = 16 $, since $ DG = 8 $, midpoint $ G $).
- $ GJ = 10.6 $: $ \frac{1}{2}\times21.2 = 10.6 $, so $ EF = 21.2 $? But diagram shows $ EF = 10.6 $. Wait, maybe the diagram has $ EF = 10.6 $ as half, so $ EF = 21.2 $? No, the diagram labels $ EF = 10.6 $. Wait, maybe the label is $ EH = 10.6 $, not $ EF $. Oh! That's the mistake. So $ EH = 10.6 $, so $ EF = 2\times EH = 21.2 $, and $ GJ = \frac{1}{2}EF = 10.6 $ (correct, matches first option). So:

- $ DE = 24 $, so $ GH = \frac{1}{2}DE = 12 $.
- $ DF = 16 $ (since $ DG = 8 $, $ G $ is midpoint), so $ JH = \frac{1}{2}DF = 8 $.
- $ EF = 21.2 $ (since $ EH = 10.6 $, $ H $ is midpoint), so $ GJ = \frac{1}{2}EF = 10.6 $.

Thus, the first option is correct.

## Step2: Verify the options
- Option 1: $ JH = 8 $, $ GH = 12 $, $ GJ = 10.6 $. Matches the midsegment lengths calculated ( $ GH = 12 $ from $ DE = 24 $, $ JH = 8 $ from $ DF = 16 $, $ GJ = 10.6 $ from $ EF = 21.2 $ ).
- Other options:
  - Option 2: $ JH = 2 $, $ GH = 6 $, $ GJ = 2.8 $ – too small.
  - Option 3: $ JH = 16 $, $ GH = 48 $, $ GJ = 21.2 $ – too big (double the sides, not half).
  - Option 4: $ JH = 6 $, $ GH = 8 $, $ GJ = 3.2 $ – doesn't match.

# Answer:
A. $ JH = 8 $, $ GH = 12 $ and $ GJ = 10.6 $

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QUESTION IMAGE

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Explanation:

Step1: Recall Midsegment Theorem

Answer: