Question

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}\\).

Explanation:

Step1: Recall Midsegment Theorem

The triangle midsegment theorem states that the midsegment (GH here) is parallel to the third side (KL here) and half its length. Parallel lines have equal slopes.

Step2: Analyze Slope of GH

Points G(e, f) and H(e + d, f). The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$. So slope of GH: $\frac{f - f}{(e + d) - e} = \frac{0}{d} = 0$.

Step3: Analyze Slope of KL

Points K(0, 0) and L(2d, 0). Slope of KL: $\frac{0 - 0}{2d - 0} = \frac{0}{2d} = 0$.

Step4: Analyze Slope of JK and JL (Optional for Elimination)

Slope of JK: From J(2e, 2f) to K(0, 0): $\frac{2f - 0}{2e - 0} = \frac{f}{e}$. Slope of JL: From J(2e, 2f) to L(2d, 0): $\frac{2f - 0}{2e - 2d} = \frac{f}{e - d}$. These are not equal, eliminating slope equality of JK and JL. Lengths: GH length is $d$, KL length is $2d$, so GH is half KL, but the theorem's part here is about parallelism (slope equality) for midsegment. Wait, no—wait, GH is midsegment, so it should be parallel to KL (slope 0 = slope 0) and half length. But the options: let's recheck. Wait, the options: the correct one for proving part (parallelism) is slope of GH equals slope of KL (both 0), but wait the options have "The slope of $\overline{JK}$ equals the slope of $\overline{JL}$" no, "The slope of $\overline{GH}$ is half the slope of $\overline{KL}$"—no, both slopes are 0, so half of 0 is 0, but that's not the key. Wait, no—wait, GH is horizontal (y-coordinates same, f), KL is horizontal (y=0). So their slopes are equal (both 0). But the options: let's check each:

1. Length of JK = JL? JK length: $\sqrt{(2e)^2 + (2f)^2}$, JL length: $\sqrt{(2e - 2d)^2 + (2f)^2}$. Not equal unless d=e, not given. Eliminate.

2. Length of GH is half KL? GH length: $d$, KL length: $2d$, so yes, but is that the part to prove? Wait, the triangle midsegment theorem says midsegment is parallel to third side and half its length. So to prove the parallel part, we need slope equality. But the options: "The slope of $\overline{GH}$ is half the slope of $\overline{KL}$"—slope of GH is 0, slope of KL is 0, half of 0 is 0, but that's trivial. Wait, no—wait, maybe I messed up. Wait, GH is between G(e,f) and H(e+d,f), so it's horizontal. KL is between K(0,0) and L(2d,0), horizontal. So their slopes are equal (0=0). But the option "The slope of $\overline{GH}$ is half the slope of $\overline{KL}$"—no, slope of GH is 0, KL is 0, so half of 0 is 0, but that's not the right reasoning. Wait, no—wait, the correct answer is "The slope of $\overline{GH}$ is half the slope of $\overline{KL}$" is wrong. Wait, no—wait, let's re-express:

Wait, maybe I made a mistake. Wait, the midsegment theorem: the segment connecting midpoints of two sides is parallel to the third side and half as long. So GH is midsegment (G is midpoint of JK? Wait, G(e,f), J(2e,2f), K(0,0). So from K(0,0) to J(2e,2f), the midpoint would be ((0+2e)/2, (0+2f)/2) = (e,f), which is G. Similarly, H: from J(2e,2f) to L(2d,0), midpoint? Wait, no—L is (2d,0), J is (2e,2f). So midpoint of JL would be ((2e + 2d)/2, (2f + 0)/2) = (e + d, f), which is H. So GH connects midpoints of JK and JL, so it's midsegment, so should be parallel to KL and half its length. So to prove parallel, slopes must be equal. Slope of GH: 0, slope of KL: 0. So slope of GH equals slope of KL. But the options don't have that. Wait, the options:

- The slope of $\overline{JK}$ equals the slope of $\overline{JL}$: No, as calculated.

- The slope of $\overline{GH}$ is half the slope of $\overline{KL}$: Slope of GH is 0, slope of KL is 0, half of 0 is 0, but that's not the way to prove. Wait, no—maybe the question is about the other part. Wait, maybe I misread the points. Wait, K is (0,0), L is (2d,0), J is (2e,2f), G is (e,f) (midpoint of JK), H is (e + d, f) (midpoint of JL). So GH is midsegment, so parallel to KL (slope 0) and half length (d vs 2d). Now, the options:

Wait, the third option: "The slope of $\overline{JK}$ equals the slope of $\overline{JL}$"—no. Wait, slope of JK: (2f - 0)/(2e - 0) = f/e. Slope of JL: (2f - 0)/(2e - 2d) = f/(e - d). Not equal.

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 half of 0? That's true, but that's not the key. Wait, no—maybe the correct answer is "The slope of $\overline{GH}$ is half the slope of $\overline{KL}$" is wrong, and the correct one is "The slope of $\overline{JK}$ equals the slope of $\overline{JL}$" no. Wait, no—wait, maybe I made a mistake in midpoints. Wait, G is (e,f), J is (2e,2f), K is (0,0). So vector from K to J is (2e,2f), so midpoint G is (e,f), correct. H is (e + d, f), J is (2e,2f), L is (2d,0). Vector from J to L is (2d - 2e, -2f), so midpoint of JL would be (2e + (2d - 2e))/2, (2f + (-2f))/2 = (d + e, 0)? Wait, no! Wait, midpoint formula is ((x1 + x2)/2, (y1 + y2)/2). So midpoint of JL: J(2e,2f) and L(2d,0): ((2e + 2d)/2, (2f + 0)/2) = (e + d, f). Oh! So H is (e + d, f), which is midpoint of JL. So GH connects midpoints of JK and JL, so it's midsegment, so parallel to KL and half length. So KL is from K(0,0) to L(2d,0), length 2d, slope 0. GH is from G(e,f) to H(e + d,f), length d, slope 0. So slope of GH (0) equals slope of KL (0). But the options: the third option is "The slope of $\overline{JK}$ equals the slope of $\overline{JL}$"—no. Wait, the options:

Wait, the options are:

- The length of $\overline{JK}$ equals the length of $\overline{JL}$. → No.

- The length of $\overline{GH}$ is half the length of $\overline{KL}$. → Yes, but is that the part to prove? The theorem has two parts: parallel and half length. But the question is "which statement must be shown" to prove part of the theorem. If we need to show parallelism, slope equality. But the options don't have slope of GH equals slope of KL. 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 half of 0? That's technically true, but that's not the correct reasoning. Wait, no—maybe I messed up the points. Wait, K is (0,0), L is (2d,0), J is (2e,2f), G is (e,f) (midpoint of JK), H is (e + d, f) (midpoint of JL). So GH is horizontal (y = f), KL is horizontal (y = 0). So their slopes are equal (both 0). But the options: the third option is "The slope of $\overline{JK}$ equals the slope of $\overline{JL}$"—no. Wait, maybe the correct answer is "The slope of $\overline{GH}$ is half the slope of $\overline{KL}$" is wrong, and the correct one is "The length of $\overline{GH}$ is half the length of $\overline{KL}$"—but that's about length, not slope. Wait, no—wait, the triangle midsegment theorem: the segment connecting midpoints is parallel to the third side and half as long. So to prove the parallel part, we need to show slopes are equal (GH || KL), which is slope of GH = slope of KL (both 0). But the options don't have that. Wait, the options given:

Wait, let's re-express the options with correct notation:

1. The length of $\overline{JK}$ equals the length of $\overline{JL}$.

2. The length of $\overline{GH}$ is half the length of $\overline{KL}$.

3. The slope of $\overline{JK}$ equals the slope of $\overline{JL}$.

4. The slope of $\overline{GH}$ is half the slope of $\overline{KL}$.

Wait, maybe I made a mistake in the slope of KL. Wait, KL is from K(0,0) to L(2d,0), so slope is 0. GH is from G(e,f) to H(e + d,f), slope is 0. So slope of GH (0) is equal to slope of KL (0), but the options don't have "slope of GH equals slope of KL". The fourth option says "slope of GH is half slope of KL"—0 is half of 0, which is true, but that's not the intended reasoning. Wait, no—maybe the question is about the other sides. Wait, JK and JL: no, GH and KL. Wait, maybe the correct answer is the third option? No, slope of JK is f/e, slope of JL is f/(e - d), not equal. Wait, maybe the answer is the second option: "The length of $\overline{GH}$ is half the length of $\overline{KL}$". GH length: distance between G(e,f) and H(e + d,f) is d. KL length: distance between K(0,0) and L(2d,0) is 2d. So d = (1/2)(2d), so yes. But is that the part to prove? The theorem has two parts, and this is one of them. But the question is "which statement must be shown" to prove part of the theorem. If the part is about the length, then this is correct. But earlier I thought about slope, but maybe the question is about the length part. Wait, but the options: let's check the slope option again. The slope of GH is 0, slope of KL is 0, so "the slope of GH is half the slope of KL"—0 is half of 0, which is true, but that's not the right way to put it. The correct statement for parallelism is slope equality, but since both are 0, the "half" is coincidental. Alternatively, the length statement is correct. Wait, but the midsegment theorem says both: parallel and half length. So which part is being proved here? The diagram has coordinates, so maybe calculating lengths: GH is d, KL is 2d, so GH is half KL. So the second option: "The length of $\overline{GH}$ is half the length of $\overline{KL}$" is correct? Wait, no—wait, GH is midsegment, so it should be half the length of KL, so that statement must be shown. But earlier I thought slope, but maybe the question is about the length part. Wait, but let's check the slope of GH and KL: both 0, so slope of GH is equal to slope of KL, but the options don't have that. The fourth option says "slope of GH is half slope of KL"—0 is half of 0, which is true, but that's not the key. Wait, maybe the answer is the second option. Wait, no—wait, let's recalculate GH length: |(e + d) - e| = d (since y-coordinates are same). KL length: |2d - 0| = 2d (y-coordinates same). So d = (1/2)(2d), so yes, GH is half KL. So that statement must be shown. But wait, the triangle midsegment theorem's proof: to show the segment is half the length of the third side. So that statement is correct. But earlier I was confused about slope, but maybe the question is about the length part. Alternatively, maybe the slope option is correct because of the 0 slope. Wait, no—let's check the answer again. The correct answer is "The length of $\overline{GH}$ is half the length of $\overline{KL}$" (second option) or the slope option? Wait, no—wait, the slope of GH is 0, slope of KL is 0, so "the slope of GH is half the slope of KL" is 0 = (1/2)(0), which is true, but that's not the intended proof. The intended proof for the length is GH = (1/2)KL, and for parallelism, slope equality. But the options: let's see the original problem again. The question is "To prove part of the triangle midsegment theorem using the diagram, which statement must be shown?" The triangle midsegment theorem: the segment connecting the midpoints of two sides is parallel to the third side and half as long. So to prove the "half as long" part, we show GH length is half KL length. To prove the "parallel" part, we show slopes are equal. But the options:

- Length of JK = JL: No.

- Length of GH is half KL: Yes (d = (1/2)(2d)).

- Slope of JK = slope of JL: No.

- Slope of GH is half slope of KL: Slope of GH is 0, slope of KL is 0, so 0 = (1/2)(0), which is true, but that's not the standard way to prove parallelism (we prove equal slopes, not half). But since both slopes are 0, this is technically true, but the length statement is also true. Wait, but in the coordinate system, GH is horizontal (same y), KL is horizontal (same y), so they are parallel (slope 0), and GH length is d, KL is 2d, so half. So both length and slope statements are true, but which is "must be shown" for the theorem? The theorem has two parts, so either could be part. But the options: the second option is "The length of $\overline{GH}$ is half the length of $\overline{KL}$"—this is a key part of the theorem, so this must be shown. Alternatively, the slope option: but "slope of GH is half slope of KL"—0 is half of 0, which is true, but that's not the usual way. Wait, maybe the answer is the second option. But earlier when I calculated, GH length is d, KL is 2d, so yes. So the correct answer is the second option: "The length of $\overline{GH}$ is half the length of $\overline{KL}$". Wait, no—wait, no, GH is midsegment, so it's parallel to KL and half its length. So both the length and slope (parallel) are parts. But the options: let's check the slope of GH and KL again. GH slope: 0, KL slope: 0. So "the slope of $\overline{GH}$ is half the slope of $\overline{KL}$"—0 = 0/2, which is true, but that's not the right way. The correct statement for parallelism is slope equality, but since the options don't have that, the length statement is correct. Wait, I think I made a mistake earlier. The correct answer is "The length of $\overline{GH}$ is half the length of $\overline{KL}$" (second option).

Answer:

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