Since RS is a midsegment parallel to AC, that means R is the midpoint of AB and S is the midpoint of BC. The midpoint formula is:
[tex](\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})[/tex]. Using the coordinates of A and B, we have:
[tex]R=(\frac{5+2}{2},\frac{8+3}{2})
\\R=(\frac{7}{2},\frac{11}{2})
\\R=(3.5, 5.5)[/tex]. Similarly, S is the midpoint of BC:
[tex]S=(\frac{5+8}{2},\frac{8+3}{2})
\\S=(\frac{13}{2},\frac{11}{2})
\\S=(6.5, 5.5)[/tex].
Since ST is a midsegment parallel to AB, then T must be a midpoint of AC:
[tex]T=(\frac{2+8}{2},\frac{3+3}{2})
\\T=(\frac{10}{2},\frac{6}{2})
\\T=(5,3)[/tex].
Now that we have the coordinates of each point we can find the length of each segment using the distance formula:
[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
For ST:
[tex]d=\sqrt{(6.5-5)^2+(5.5-3)^2}
\\=\sqrt{(1.5)^2+(2.5)^2}
\\=\sqrt{2.25+6.25
\\=\sqrt{8.5}=2.9 \neq 4[/tex]
For RT:
[tex]d=\sqrt{(3.5-5)^2+(5.5-3)^2}
\\=\sqrt{(-1.5)^2+(2.5)^2}
\\=\sqrt{2.25+6.25}
\\=\sqrt{8.5}=2.9 \neq 5[/tex]
For RS:
[tex]d=\sqrt{(3.5-6.5)^2+(5.5-5.5)^2}
\\=\sqrt{(-3)^2+(0)^2}
\\=\sqrt{9+0}=\sqrt{9}=3[/tex]
