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: coordinate plane with points k(0, 0), l(2d, 0), j(2e, 2f), g(e, f), h(e + d, f))

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# Brief Explanations:
The Triangle Midsegment Theorem states that the midsegment (here, $ \overline{GH} $) of a triangle is parallel to the third side (here, $ \overline{KL} $) and half its length. For parallel lines, their slopes are equal. Let's analyze each option:

- Option 1: $ \overline{JK} $ and $ \overline{JL} $ are sides from the same vertex, not related to midsegment parallelism.
- Option 2: The theorem says the midsegment is half the third side’s length, but to prove the theorem (using coordinates), we first show parallelism (equal slopes) or use midpoints. However, the key for the proof step here is about slopes for parallelism. Wait, no—wait, $ G $ and $ H $: $ 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) $: $ L(2d,0) $, so 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, so it should be parallel to $ KL $ (slope 0, since $ K(0,0) $, $ L(2d,0) $, slope $ \frac{0 - 0}{2d - 0}=0 $). $ GH $ has points $ G(e,f) $, $ H(e + d,f) $, slope $ \frac{f - f}{(e + d) - e}=0 $, so slope of $ GH $ equals slope of $ KL $ (both 0). But the options: let's check each:

- "The length of $ \overline{JK} $ equals $ \overline{JL} $": No, $ JK $ length is $ \sqrt{(2e)^2 + (2f)^2} $, $ JL $ is $ \sqrt{(2d - 2e)^2 + (0 - 2f)^2} $, not equal.
- "The length of $ \overline{GH} $ is half $ \overline{KL} $": $ GH $ length is $ d $, $ KL $ length is $ 2d $, so this is true, but is this the "part of the proof" needed? Wait, the theorem’s proof often involves showing midpoints first, then parallelism (equal slopes) or length. But the options: the third option: "The slope of $ \overline{JK} $ equals slope of $ \overline{JL} $": $ JK $ slope is $ \frac{2f - 0}{2e - 0}=\frac{f}{e} $, $ JL $ slope is $ \frac{0 - 2f}{2d - 2e}=\frac{-f}{d - e} $, not equal. Fourth option: "Slope of $ GH $ is half slope of $ KL $": $ KL $ slope is 0, $ GH $ slope is 0, so 0 is not half of 0 (trivially, but the statement is wrong). Wait, no—wait, the correct approach: the midsegment $ GH $ should be parallel to $ KL $, so their slopes must be equal. But $ KL $ is horizontal (slope 0), $ GH $ is horizontal (slope 0). But the options: the third option is "The slope of $ \overline{JK} $ equals the slope of $ \overline{JL} $"—no. Wait, maybe I misread. Wait, $ G $ and $ H $ are midpoints: $ G $ is midpoint of $ JK $, $ H $ is midpoint of $ JL $. So by midsegment theorem, $ GH \parallel KL $ and $ GH = \frac{1}{2}KL $. To prove parallelism, slopes must be equal. $ KL $ has slope $ \frac{0 - 0}{2d - 0}=0 $. $ GH $ has slope $ \frac{f - f}{(e + d) - e}=0 $, so slope of $ GH $ equals slope of $ KL $, but that's not an option. Wait, the options: let's re-express:

Wait, the options are:

1. Length $ JK = JL $: No.
2. Length $ GH = \frac{1}{2}KL $: $ GH $ length is $ (e + d) - e = d $, $ KL $ length is $ 2d - 0 = 2d $, so $ GH = \frac{1}{2}KL $. But is this the "part of the proof" needed? Wait, the theorem’s proof: to show $ GH $ is midsegment, we first show $ G $ and $ H $ are midpoints, then show $ GH \parallel KL $ (equal slopes) and $ GH = \frac{1}{2}KL $. But the question is "which statement must be shown" to prove part of the theorem. Let's check the slopes of $ JK $ and $ JL $: no. Wait, maybe the key is that $ G $ and $ H $ are midpoints, so $ GH $ is midsegment, so it should be parallel to $ KL $, meaning their slopes are equal. But $ KL $ is horizontal, $ GH $ is horizontal, so slope 0. But the third option: "The slope of $ \overline{JK} $ equals the slope of $ \overline{JL} $"—no. Wait, maybe I made a mistake. Wait, $ K(0,0) $, $ L(2d,0) $, so $ KL $ is along x-axis. $ G(e,f) $, $ H(e + d,f) $, so $ GH $ is horizontal (same y-coordinate), so slope 0, same as $ KL $. But the options: the third option is "The slope of $ \overline{JK} $ equals the slope of $ \overline{JL} $": $ JK $ is from $ (0,0) $ to $ (2e,2f) $, slope $ \frac{2f}{2e}=\frac{f}{e} $. $ JL $ is from $ (2e,2f) $ to $ (2d,0) $, slope $ \frac{0 - 2f}{2d - 2e}=\frac{-f}{d - e} $. Not equal. The fourth option: "Slope of $ GH $ is half slope of $ KL $": $ KL $ slope is 0, $ GH $ slope is 0, so 0 is not half of 0 (the statement is false). The second option: "Length of $ GH $ is half length of $ KL $": $ GH $ length is $ d $, $ KL $ length is $ 2d $, so this is true. But is this the "part of the proof" needed? Wait, the theorem says midsegment is half the third side and parallel. To prove the theorem, we can show midpoints, then show length and slope. But the options: the correct answer is the second option? Wait, no—wait, maybe I messed up. Wait, the midsegment theorem: the segment connecting midpoints of two sides is parallel to the third side and half its length. So to prove part of it (using the diagram), we need to show either the length or the slope. Let's check the coordinates:

- $ GH $ length: distance between $ (e,f) $ and $ (e + d,f) $ is $ d $.
- $ KL $ length: distance between $ (0,0) $ and $ (2d,0) $ is $ 2d $.
So $ GH = \frac{1}{2}KL $, so the second option is correct. Wait, but earlier I thought about slope, but the options: let's re-express all options:

1. Length $ JK = JL $: $ JK = \sqrt{(2e)^2 + (2f)^2} $, $ JL = \sqrt{(2d - 2e)^2 + (0 - 2f)^2} $. Not equal.
2. Length $ GH = \frac{1}{2}KL $: $ GH = d $, $ KL = 2d $, so $ GH = \frac{1}{2}KL $. Correct.
3. Slope $ JK = $ slope $ JL $: $ \frac{2f}{2e} = \frac{-2f}{2d - 2e} $ → $ \frac{f}{e} = \frac{-f}{d - e} $ → $ d - e = -e $ → $ d = 0 $, which is not general. So false.
4. Slope $ GH = \frac{1}{2} $ slope $ KL $: slope $ KL = 0 $, slope $ GH = 0 $, so $ 0 = \frac{1}{2}(0) $, which is true, but the statement says "half the slope", which is technically true, but the theorem is about length and parallelism (equal slope), not half slope. Also, the key part of the theorem is that the midsegment is half the length of the third side, so the second option is the correct part to show.

Wait, no—wait, the midsegment theorem’s proof: we first show that $ G $ and $ H $ are midpoints (which they are, by coordinates: $ G $ is midpoint of $ JK $, $ H $ is midpoint of $ JL $). Then, to prove $ GH \parallel KL $ and $ GH = \frac{1}{2}KL $. The slope of $ GH $ is 0 (horizontal), slope of $ KL $ is 0 (horizontal), so they are parallel (equal slopes). Then, length: $ GH = d $, $ KL = 2d $, so $ GH = \frac{1}{2}KL $. So both slope equality and length relation are part of the theorem. But the options: the second option is "The length of $ \overline{GH} $ is half the length of $ \overline{KL} $", which is a key part of the theorem. The third option is about slopes of $ JK $ and $ JL $, which are not related. The fourth option is incorrect (slope of $ GH $ is equal to slope of $ KL $, not half). So the correct answer is the second option.

# Answer:
The length of $ \overline{GH} $ is half the length of $ \overline{KL} $ (the second option).

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