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 { j k } \\) equals the length of \\( \overline { j l } \\).
the length of \\( \overline { g h } \\) is half the length of \\( \overline { k l } \\).
the slope of \\( \overline { j k } \\) equals the slope of \\( \overline { j l } \\).
the slope of \\( \overline { g h } \\) is half the slope of \\( \overline { k l } \\).

Explanation:

Step1: Recall the Triangle Midsegment Theorem

The Triangle Midsegment Theorem states that the midsegment (a segment connecting the mid - points of two sides of a triangle) is parallel to the third side and half its length. In this case, \(\overline{GH}\) is a mid - segment (since \(G\) and \(H\) are likely mid - points, given the coordinate pattern).

Step2: Analyze each option

• Option 1: The length of \(\overline{JK}\) equals the length of \(\overline{JL}\). This is about two sides of the triangle \( \triangle JKL\) and has no relation to the mid - segment theorem.
• Option 2: Calculate the lengths.

The length of \(\overline{GH}\) using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For \(G(e,f)\) and \(H(e + d,f)\), \(d_{GH}=\sqrt{(e + d-e)^2+(f - f)^2}=d\). For \(K(0,0)\) and \(L(2d,0)\), \(d_{KL}=\sqrt{(2d-0)^2+(0 - 0)^2}=2d\). So \(d_{GH}=\frac{1}{2}d_{KL}\).

• Option 3: The slope of \(\overline{JK}\) and \(\overline{JL}\). The slope of \(\overline{JK}\) with \(J(2e,2f)\) and \(K(0,0)\) is \(m_{JK}=\frac{2f-0}{2e - 0}=\frac{f}{e}\). The slope of \(\overline{JL}\) with \(J(2e,2f)\) and \(L(2d,0)\) is \(m_{JL}=\frac{2f-0}{2e-2d}=\frac{f}{e - d}\) (assuming \(e

eq d\)). These slopes are not equal.

• Option 4: The slope of \(\overline{GH}\) (since \(y\) - values are the same for \(G\) and \(H\), \(m_{GH}=0\)). The slope of \(\overline{KL}\) (since \(y\) - values are the same for \(K\) and \(L\), \(m_{KL}=0\)). But \(0

eq\frac{1}{2}\times0\) in a non - meaningful way for the mid - segment theorem.

Answer:

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