Answer:
See below.
Explanation:
30.
E(1, 2), F(5, 6), G(3, -2)
a)
H, midpoint of segment EG
H( (1 + 3)/2, (2 + (-2))/2) = H(2, 0)
J, midpoint of segment FG
J( (5 + 3)/2, (6 + (-2))/2) = J(4, 2)
b)
Show that segment HJ is parallel to segment EF.
Parallel lines have equal slopes. We find the slopes of the two segments using the endpoints and show that the slopes are equal proving the segments are parallel.
slope of segment HJ:
slope HJ = (2 - 0)/(4 - 2) = 2/2 = 1
slope of segment EF:
slope EF = (6 - 2)/(5 - 1) = 4/4 = 1
slope HJ = slope EF = 1
Segments HJ and EF are parallel since their slopes are equal.
c)
Show HJ = (1/2)EF
We use the distance formula to find the lengths of segments HJ and EF.
length of HJ = sqrt[(4 - 2)^2 + (2 - 0)^2] = sqrt(4 + 4) = sqrt(8) = 2sqrt(2)
length of EF = sqrt[(5 - 1)^2 + (6 - 2)^2] = sqrt(16 + 16) = sqrt(32) = 4sqrt(2)
HJ = (1/2)EF
Substitute the lengths we found just above for HJ and EF:
2sqrt(2) = (1/2)[4sqrt(2)]
2sqrt(2) = 2sqrt(2)
The two sides are equal, so we have proved that HJ = (1/2)EF.