explaining part of the proof of the triangle midsegment theorem
to prove part of the triangle midsegment theorem using the diagram, which statement must be shown?
○ the length of $overline{jk}$ equals the length of $overline{jl}$.
○ the length of $overline{gh}$ is half the length of $overline{kl}$.
○ the slope of $overline{jk}$ equals the slope of $overline{jl}$.
○ the slope of $overline{gh}$ is half the slope of $overline{kl}$.
diagram: points $k(0, 0)$, $l(2d, 0)$, $j(2e, 2f)$, $g(e, f)$, $h(e + d, f)$ on a coordinate plane with $x$- and $y$-axes.

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# Brief Explanations:
The Triangle Midsegment Theorem states that the midsegment of a triangle is parallel to the third side and half its length. To prove parallelism, we check if slopes are equal.
- For $ \overline{GH} $: Points $ G(e,f) $ and $ H(e + d,f) $. Slope of $ \overline{GH} $ is $ \frac{f - f}{(e + d)-e}=\frac{0}{d} = 0 $.
- For $ \overline{KL} $: Points $ K(0,0) $ and $ L(2d,0) $. Slope of $ \overline{KL} $ is $ \frac{0 - 0}{2d-0}=\frac{0}{2d}=0 $.
- For $ \overline{JK} $ (points $ J(2e,2f) $ and $ K(0,0) $): Slope is $ \frac{2f - 0}{2e - 0}=\frac{f}{e} $.
- For $ \overline{JL} $ (points $ J(2e,2f) $ and $ L(2d,0) $): Slope is $ \frac{2f - 0}{2e - 2d}=\frac{f}{e - d} $, which is not equal to slope of $ \overline{JK} $ (so third option is wrong).
- Length of $ \overline{GH} $: $ d $, length of $ \overline{KL} $: $ 2d $, so $ \overline{GH} $ is half of $ \overline{KL} $, but the theorem's proof for parallelism (part of the proof) needs slope equality. Wait, no—wait, the question is "to prove part of the triangle midsegment theorem". The midsegment $ GH $ should be parallel to $ KL $ (so same slope) and half its length. But let's re - evaluate the options:
    - First option: Length of $ \overline{JK} $ and $ \overline{JL} $: $ JK=\sqrt{(2e)^2+(2f)^2} $, $ JL=\sqrt{(2e - 2d)^2+(2f)^2} $, not equal. So first option wrong.
    - Second option: Length of $ \overline{GH}=d $, $ \overline{KL}=2d $, so $ GH=\frac{1}{2}KL $, but is this the "part" to show? Wait, the midsegment theorem has two parts: parallel and half - length. But the diagram has $ G $ and $ H $ as midpoints? Wait, $ G(e,f) $ is midpoint of $ JK $ (since $ J(2e,2f) $ and $ K(0,0) $, midpoint is $ (\frac{2e + 0}{2},\frac{2f+0}{2})=(e,f) $), and $ H(e + d,f) $: midpoint of $ JL $? $ J(2e,2f) $, $ L(2d,0) $, midpoint is $ (\frac{2e + 2d}{2},\frac{2f+0}{2})=(e + d,f) $. So $ GH $ is midsegment. To prove part of the theorem (parallelism or length), let's check slopes. Slope of $ GH $ is 0, slope of $ KL $ is 0, so they are parallel. But the third option: slope of $ JK $ and $ JL $. Slope of $ JK=\frac{2f-0}{2e - 0}=\frac{f}{e} $, slope of $ JL=\frac{2f-0}{2e - 2d}=\frac{f}{e - d} $, not equal. Wait, no—wait the third option says "the slope of $ \overline{JK} $ equals the slope of $ \overline{JL} $"—no, that's not true. Wait, maybe I made a mistake. Wait, $ G $ is midpoint of $ JK $, $ H $ is midpoint of $ JL $, so $ GH \parallel KL $ (by midsegment theorem). So to prove parallelism, we need to show that slope of $ GH $ equals slope of $ KL $, but that's not an option. Wait, the options:
    - Option 3: "The slope of $ \overline{JK} $ equals the slope of $ \overline{JL} $"—no, as we saw. Wait, no—wait $ K(0,0) $, $ L(2d,0) $, so $ KL $ is horizontal. $ G(e,f) $, $ H(e + d,f) $, so $ GH $ is horizontal (slope 0). $ JK $: from $ (0,0) $ to $ (2e,2f) $, slope $ \frac{2f}{2e}=\frac{f}{e} $. $ JL $: from $ (2e,2f) $ to $ (2d,0) $, slope $ \frac{0 - 2f}{2d - 2e}=\frac{-f}{d - e}=\frac{f}{e - d} $. These are not equal. Wait, maybe the question is about showing that $ GH $ is parallel to $ KL $, which would mean their slopes are equal. But the option about $ GH $ and $ KL $ slope is not there. Wait, the fourth option: "The slope of $ \overline{GH} $ is half the slope of $ \overline{KL} $". Slope of $ GH $ is 0, slope of $ KL $ is 0, so 0 is not half of 0 (trivially, but the statement is wrong). Wait, the second option: "The length of $ \overline{GH} $ is half the length of $ \overline{KL} $". Length of $ GH $: distance between $ (e,f) $ and $ (e + d,f) $ is $ d $. Length of $ KL $: distance between $ (0,0) $ and $ (2d,0) $ is $ 2d $. So $ GH=\frac{1}{2}KL $. But the midsegment theorem has two parts: parallel and half - length. But the question is "to prove part of the triangle midsegment theorem". However, let's re - check the options. Wait, maybe I misread the points. $ K(0,0) $, $ L(2d,0) $, $ J(2e,2f) $, $ G(e,f) $ (midpoint of $ JK $), $ H(e + d,f) $ (midpoint of $ JL $). So $ GH $ is midsegment. The midsegment theorem says $ GH\parallel KL $ and $ GH=\frac{1}{2}KL $. To prove the parallel part, we can show that the slope of $ GH $ equals the slope of $ KL $ (both 0). But the option about $ GH $ and $ KL $ slope is not there. The third option: slope of $ JK $ and $ JL $. Wait, no—maybe the question is about showing that $ G $ and $ H $ are midpoints? No, the options are about lengths or slopes. Wait, let's recalculate slopes:
    - Slope of $ JK $: $ m_{JK}=\frac{2f - 0}{2e - 0}=\frac{f}{e} $
    - Slope of $ JL $: $ m_{JL}=\frac{0 - 2f}{2d - 2e}=\frac{-f}{d - e}=\frac{f}{e - d} $
    - Slope of $ GH $: $ m_{GH}=\frac{f - f}{(e + d)-e}=0 $
    - Slope of $ KL $: $ m_{KL}=\frac{0 - 0}{2d - 0}=0 $
    Now, the third option says "The slope of $ \overline{JK} $ equals the slope of $ \overline{JL} $"—no. The fourth option: "The slope of $ \overline{GH} $ is half the slope of $ \overline{KL} $"—0 is not half of 0 (the statement is incorrect). The first option: length of $ JK $ and $ JL $: $ JK = \sqrt{(2e)^2+(2f)^2}=2\sqrt{e^{2}+f^{2}} $, $ JL=\sqrt{(2e - 2d)^2+(2f)^2}=2\sqrt{(e - d)^{2}+f^{2}} $, not equal. The second option: length of $ GH = d $, length of $ KL = 2d $, so $ GH=\frac{1}{2}KL $. But the midsegment theorem's proof: to show that $ GH $ is parallel to $ KL $ (slopes equal) and $ GH=\frac{1}{2}KL $. But among the options, the correct one related to the proof (part of it) is that the slope of $ GH $ equals slope of $ KL $, but that's not an option. Wait, no—wait the third option: "The slope of $ \overline{JK} $ equals the slope of $ \overline{JL} $"—no. Wait, maybe I made a mistake in the midpoints. Wait, $ G(e,f) $: if $ K(0,0) $ and $ J(2e,2f) $, then midpoint of $ JK $ is $ (e,f) $, correct. $ H(e + d,f) $: midpoint of $ JL $: $ J(2e,2f) $, $ L(2d,0) $, midpoint is $ (e + d,f) $, correct. So $ GH $ is midsegment. Now, the triangle midsegment theorem: the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half as long. So to prove part of it (the parallel part), we can show that the slope of $ GH $ equals the slope of $ KL $ (both 0). But the option about $ GH $ and $ KL $ slope is not there. Wait, the third option: slope of $ JK $ and $ JL $. Wait, no—maybe the question is not about $ GH $ and $ KL $ but about $ JK $ and $ JL $? No, that doesn't make sense. Wait, let's re - examine the options again. The third option: "The slope of $ \overline{JK} $ equals the slope of $ \overline{JL} $"—no. The second option: "The length of $ \overline{GH} $ is half the length of $ \overline{KL} $"—this is a part of the midsegment theorem (the length part). The first option is wrong, the third is wrong, the fourth is wrong. So the correct option is the second one? Wait, no—wait the midsegment theorem's proof: we can use coordinate geometry. Let's calculate lengths:
    - Length of $ GH $: $ \sqrt{(e + d - e)^2+(f - f)^2}=\sqrt{d^{2}} = d $
    - Length of $ KL $: $ \sqrt{(2d - 0)^2+(0 - 0)^2}=2d $
    So $ GH=\frac{1}{2}KL $, which is the second option. But earlier I thought the parallel part, but maybe the "part" here is the length part. So the correct statement to show is that the length of $ \overline{GH} $ is half the length of $ \overline{KL} $. Wait, but also, the slope of $ GH $ and $ KL $ are both 0, so they are parallel. But the options: the third option is about $ JK $ and $ JL $ slopes, which are not equal. The second option is about the length, which is a part of the theorem. So the correct answer is the second option: "The length of $ \overline{GH} $ is half the length of $ \overline{KL} $". Wait, no—wait I think I made a mistake. The midsegment theorem says that the midsegment is parallel to the third side and half its length. So to prove part of the theorem (either parallel or length), among the options, the second option is correct because $ GH $ is the midsegment, so its length should be half of $ KL $'s length. The slope of $ GH $ and $ KL $ are both 0, so they are parallel, but the option about their slopes is not given. The third option is incorrect because slopes of $ JK $ and $ JL $ are not equal. The first option is incorrect. The fourth option is incorrect. So the correct option is the second one.
# Answer:
The length of $\overline{GH}$ is half the length of $\overline{KL}$ (the second option).

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