Question

triangle similarity
xplaining 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 slope of \\(\overline{jk}\\) equals the slope of \\(\overline{jl}\\).
the slope of \\(\overline{gh}\\) is half the slope of \\(\overline{kl}\\).
the length of \\(\overline{gh}\\) is half the length of \\(\overline{kl}\\).
the length of \\(\overline{jk}\\) equals the length of \\(\overline{jl}\\).

Explanation:

Step1: Recall Midsegment Theorem

The Triangle Midsegment Theorem states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half as long. Here, \( G \) and \( H \) should be midpoints, so \( GH \) is the midsegment, and \( KL \) is the third side.

Step2: Analyze Each Option

• Option 1 (Slope of \( \overline{JK} \) = Slope of \( \overline{JL} \)): \( JK \) and \( JL \) are sides from \( J \) to \( K \) and \( J \) to \( L \), not related to midsegment parallelism to \( KL \). Eliminate.
• Option 2 (Slope of \( \overline{GH} \) is half slope of \( \overline{KL} \)): \( KL \) is on the x - axis, slope \( = 0 \). \( GH \) is horizontal (same y - coordinate), slope \( = 0 \). Slopes are equal, not half. Eliminate.
• Option 3 (Length of \( \overline{GH} \) is half length of \( \overline{KL} \)): Coordinates of \( G(e,f) \), \( H(e + d,f) \), so length \( GH=d \). Coordinates of \( K(0,0) \), \( L(2d,0) \), length \( KL = 2d \). So \( GH=\frac{1}{2}KL \), which matches the midsegment theorem.
• Option 4 (Length of \( \overline{JK} \) = Length of \( \overline{JL} \)): \( JK \) and \( JL \) lengths: \( JK=\sqrt{(2e - 0)^2+(2f - 0)^2}=\sqrt{4e^{2}+4f^{2}} \), \( JL=\sqrt{(2e - 2d)^2+(2f - 0)^2}=\sqrt{4(e - d)^{2}+4f^{2}} \). These are equal only if \( d = 0 \), not generally true. Eliminate.

Answer:

The length of \( \overline{GH} \) is half the length of \( \overline{KL} \) (the third option in the list of statements).