Answer :
The coordinates of the vertices of a triangle are E(4,5), F(16,17 and G(10, 5) Let H be the midpoint of segment "EG" and let J be the midpoint of segment "FG".
Verify the Triangle Midsegment Theorem by showing that segment "HJ" is parallel to segment "EF" and HJ = 1/2EF.
step 1
Find out the midpoint H
The formula to calculate the midpoint between two points is equal to
[tex](\frac{x1+x2}{2},\frac{y1+y2}{2})[/tex]we have
E(4,5) and G(10, -5)
substitute given coordinates
[tex]\begin{gathered} H=(\frac{4+10}{2},\frac{5-5}{2}) \\ H(7,0) \end{gathered}[/tex]step 2
Find out the midpoint J
we have
F(16,17) and G(10, -5)
substitute
[tex]\begin{gathered} J=(\frac{16+10}{2},\frac{17-5}{2}) \\ J(13,6) \end{gathered}[/tex]step 3
Find out the slope HJ
H(7,0) and J(13,6)
m=(6-0)/(13-7)
m=6/6
m=1
step 4
Find out the slope EF
we have
E(4,5), F(16,17)
m=(17-5)/(16-4)
m=12/12
m=1
step 5
Compare slope HJ and slope EF
their slopes are equal
that means
HJ and EF are parallel
step 6
Find out the distance HJ
the formula to calculate the distance between two points is equal to
[tex]d=\sqrt[]{(y2-y1)^2+(x2-x1)^2}[/tex]we have
H(7,0) and J(13,6)
substitute
[tex]\begin{gathered} HJ=\sqrt[]{(6-0)^2+(13-7)^2} \\ HJ=\sqrt[]{(6)^2+(6)^2} \\ HJ=6\sqrt[]{2} \end{gathered}[/tex]step 7
Find out the distance EF
we have
E(4,5), F(16,17)
substitute
[tex]\begin{gathered} EF=\sqrt[]{(17-5)^2+(16-4)^2} \\ EF=\sqrt[]{(12)^2+(12)^2} \\ EF=12\sqrt[]{2} \end{gathered}[/tex]step 8
Verify
HJ = 1/2EF
substitute
[tex]\begin{gathered} 6\sqrt[]{2}=\frac{1}{2}\cdot(12\sqrt[]{2}) \\ 6\sqrt[]{2}=6\sqrt[]{2} \end{gathered}[/tex]is true
that means
Triangle Midsegment Theorem was verified