The measure of segment JH and segment GH is 78 units and 47 units respectively.
The figures in the image are triangles with midsegments joining the midpoints of two sides of the triangle.
In question 7)
Smaller/midsegment = MN = 5x - 16
Larger third side = JL = 4x + 34
First, we solve for x, using the midsegment theorem:
The smaller segment = half the third side
Hence:
MN = 1/2 × JL
Plug in the values:
5x - 16 = 1/2 × ( 4x + 34 )
2( 5x - 16 ) = 4x + 34
10x - 32 = 4x + 34
10x - 4x = 34 + 32
6x = 66
x = 66/6
x = 11
Now, Segment JL will be:
Segment JL = 4x + 34
Plug in x = 11
Segment JL = 4(11) + 34
Segment JL = 44 + 34
Segment JL = 78
In question 8)
Smaller/midsegment = GH = 3x - 4
Larger third side = DF = 9x - 59
Solve for x using the midsegment theorem:
3x - 4 = 1/2 × ( 9x - 59 )
2( 3x - 4 ) = 9x - 59
6x - 8 = 9x - 59
9x - 6x = -8 + 59
3x = 51
x = 51/3
x = 17
Now, Segment GH will be:
Segment GH = 3x - 4
Plug in x = 17
Segment GH = 3(17) - 4
Segment GH = 51 - 4
Segment GH = 47 units
Therefore, segment GH measures 47 units