Question

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 a midsegment (a segment connecting the midpoints of two sides of a triangle) is parallel to the third side and half its length. In this case, $\overline{GH}$ is a midsegment of the triangle.

Step2: Analyze each option

• Option 1: The length of $\overline{JK}$ equals the length of $\overline{JL}$ is not relevant to the midsegment theorem.
• Option 2: By the midsegment theorem, for midsegment $\overline{GH}$ and side $\overline{KL}$, we need to show $GH=\frac{1}{2}KL$.
• Option 3: The slope of $\overline{JK}$ equals the slope of $\overline{JL}$ would imply collinearity (which is not the case here) and is not related to the midsegment property.
• Option 4: The slope of a line segment is calculated as $m = \frac{y_2 - y_1}{x_2 - x_1}$. For $\overline{GH}$, slope $m_{GH}=\frac{f - f}{(e + d)-e}=0$. For $\overline{KL}$, slope $m_{KL}=\frac{0 - 0}{2d-0}=0$. But the midsegment theorem is about length - relation (not slope - relation in the way described in this option).

So, the statement that must be shown is that the length of $\overline{GH}$ is half the length of $\overline{KL}$.

Answer:

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