Answer:
[tex]\text{DF: $\sqrt{82}$}[/tex].
[tex]\text{EG: $\sqrt{170}$}[/tex].
Step-by-step explanation:
The two diagonals of quadrilateral [tex]{\rm D E F G}[/tex] are [tex]{\rm DF}[/tex] and [tex]{\rm EG}[/tex]. The goal is to find the length of these two segments given the coordinates of their endpoints.
The length of a line segment between [tex](x_{0},\, y_{0})[/tex] and [tex](x_{1},\, y_{1})[/tex] is:
[tex]\displaystyle \sqrt{(x_{1} - x_{0})^{2} + (y_{1} - y_{0})^{2}}[/tex].
The length of diagonal [tex]{\rm DF}[/tex] (between [tex]\text{D: $(-3,\, 4)$}[/tex] and [tex]\text{F: $(6,\, 3)$}[/tex]) would be:
[tex]\displaystyle \sqrt{(6 - (-3))^{2} + (3 - 4)^{2}} = \sqrt{82}[/tex].
Similarly, the length of diagonal [tex]{\rm EG}[/tex] (between [tex]\text{E: $(5,\, 6)$}[/tex] and [tex]\text{F: $(-2,\, -5)$}[/tex]) would be:
[tex]\displaystyle \sqrt{((-2) - 5)^{2} + ((-5) - 6)^{2}} = \sqrt{170}[/tex].
