Question

applying the midsegment theorem
if h is the midpoint of \\(\overline{ge}\\) and j is the midpoint of \\(\overline{fe}\\), determine the following lengths.
\\(hj = \square\\)
\\(je = \square\\)
(the diagram shows a geometric figure with points g, f, e, h, j. the length of gf is \\(4x - 4\\), hj is \\(x + 3\\), je is \\(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 its length. Here, \( HJ \) is the midsegment, so \( HJ=\frac{1}{2}GF \). Thus, \( x + 3=\frac{1}{2}(4x - 4) \).

Step2: Solve for \( x \)

Multiply both sides by 2: \( 2(x + 3)=4x - 4 \)
Expand: \( 2x+6 = 4x - 4 \)
Subtract \( 2x \): \( 6 = 2x - 4 \)
Add 4: \( 10 = 2x \)
Divide by 2: \( x = 5 \).

Step3: Find \( HJ \)

Substitute \( x = 5 \) into \( HJ=x + 3 \): \( HJ=5 + 3 = 8 \).

Step4: Find \( JE \)

Since \( J \) is the midpoint of \( FE \), \( JE = JF \)? Wait, no, \( JE=x - 1 \). Substitute \( x = 5 \): \( JE=5 - 1 = 4 \). Wait, but check \( JF \): \( JF \) should be equal to \( JE \)? Wait, no, \( FE=FJ + JE \), but since \( J \) is midpoint, \( FJ = JE \). Wait, but \( HJ \) is midsegment, so \( GF = 4x - 4=16 \), \( HJ = 8 \) (half of 16, correct). Now, \( JE=x - 1=4 \), and check \( JF \): \( JF \) should be equal to \( JE \), so \( JF = 4 \), then \( FE = 8 \). But also, from the triangle, \( HJ \) is midsegment, so the triangle is \( \triangle GFE \), with midpoints \( H \) (on \( GE \)) and \( J \) (on \( FE \)), so \( HJ \parallel GF \) and \( HJ=\frac{1}{2}GF \), which holds. Now, \( JE=x - 1 \), with \( x = 5 \), \( JE = 4 \).

Answer:

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