Question

applying the midsegment theorem
if h is the midpoint of \\( \overline { g e } \\) and j is the midpoint of \\( \overline { f e } \\).
determine the following lengths.
\\( h j = \\)
\\( j e = \\)

Explanation:

Step1: Apply the Midsegment Theorem

The Midsegment Theorem states that the midsegment \( HJ\) is parallel to \( GF\) and \( HJ=\frac{1}{2}GF\). So \(x + 3=\frac{1}{2}(4x-4)\).

$$\begin{align*} x + 3&=\frac{1}{2}(4x-4)\\ x+3&=2x - 2\\ 2x-x&=3 + 2\\ x&=5 \end{align*}$$

Step2: Find the length of \( HJ\)

Substitute \(x = 5\) into the expression for \( HJ\) (\(x + 3\)). So \(HJ=5+3=8\).

Step3: Find the length of \( JE\)

Since \(J\) is the mid - point of \(FE\), \(JE=x - 1\). Substitute \(x = 5\) into \(x - 1\). So \(JE=5-1=4\).

Answer:

\(HJ = 8\), \(JE=4\)