Question

applying the midsegment theorem ty it if h is the midpoint of \\(\overline{ge}\\) and j is the midpoint of \\(\overline{fe}\\), determine the following lengths. \\(hj = \square\\) \\(je = \square\\) (diagram: points g, f, e, h, j with \\(gf = 4x - 4\\), \\(hj = x + 3\\), \\(je = x - 1\\))

Explanation:

Step1: Apply Midsegment Theorem

The 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. So, \( HJ=\frac{1}{2}GF \).
Given \( GF = 4x - 4 \) and \( HJ=x + 3 \), we have the equation:
\( x + 3=\frac{1}{2}(4x - 4) \)

Step2: Solve for \( x \)

Simplify the right - hand side: \( \frac{1}{2}(4x - 4)=2x-2 \)
So the equation becomes \( x + 3=2x - 2 \)
Subtract \( x \) from both sides: \( 3=x - 2 \)
Add 2 to both sides: \( x=5 \)

Step3: Find \( HJ \)

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

Step4: Find \( JE \)

Since \( J \) is the midpoint of \( \overline{FE} \), \( JE = JF \). And we know that \( JF=x - 1 \) (wait, no, actually, from the diagram, \( JE=x - 1 \)? Wait, no, let's re - check. Wait, since \( J \) is the midpoint, \( JE = JF \), but we can also use the value of \( x \) to find \( JE \). Wait, the length of \( JE \) is given by the expression \( x-1 \)? Wait, no, let's correct. Wait, when \( x = 5 \), \( JE=x - 1=5 - 1 = 4 \)? Wait, no, wait, let's re - examine the Midsegment Theorem. Wait, the triangle is \( \triangle GFE \), with \( H \) midpoint of \( GE \) and \( J \) midpoint of \( FE \). So \( HJ\parallel GF \) and \( HJ=\frac{1}{2}GF \). We found \( x = 5 \). Then, for \( JE \), since \( J \) is the midpoint of \( FE \), but we can also check the length of \( JE \) from the expression. Wait, the segment \( JE \) has the expression \( x - 1 \)? Wait, no, maybe I misread. Wait, the diagram shows \( JE=x - 1 \), and since \( J \) is the midpoint, but also, let's confirm with \( x = 5 \). If \( x = 5 \), then \( JE=5 - 1 = 4 \). But let's check the Midsegment again. Wait, \( GF = 4x-4=4\times5 - 4=16 \), and \( HJ=x + 3=8 \), and \( \frac{1}{2}\times16 = 8 \), which matches. Then for \( JE \), since \( J \) is the midpoint of \( FE \), and if we assume that \( JE=x - 1 \), when \( x = 5 \), \( JE = 4 \).

Answer:

\( HJ=\boldsymbol{8} \), \( JE=\boldsymbol{4} \)