Question

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 triangle jkl, k at (0,0), l at (2d, 0), j at (2e, 2f). points g(e, f) and h(e + d, f) on jk and jl respectively, connected by a horizontal segment gh.

Explanation:

Step1: Identify midsegment properties

The triangle midsegment theorem states a midsegment is parallel to the third side and half its length. $\overline{GH}$ is a midsegment if $G$ and $H$ are midpoints of $\overline{JK}$ and $\overline{JL}$ (since $G(e,f)$ is midpoint of $J(2e,2f)$ and $K(0,0)$; $H(e+d,f)$ is midpoint of $J(2e,2f)$ and $L(2d,0)$).

Step2: Check length condition

Length of $\overline{GH}$: $|(e+d)-e| = d$ (y-coordinates same). Length of $\overline{KL}$: $|2d - 0| = 2d$. So $\overline{GH} = \frac{1}{2}\overline{KL}$.

Step3: Evaluate options

Only the option "The length of $\overline{GH}$ is half the length of $\overline{KL}$" matches the midsegment length property.

Answer:

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