Answer:
Option D. 90°
Step-by-step explanation:
we have
[tex]D(5, 7),E(4, 3),F(8, 2)[/tex]
Plot the vertices
see the attached figure
the formula to calculate the distance between two points is equal to
[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]
step 1
Find the distance DE
[tex]D(5, 7),E(4, 3)[/tex]
substitute
[tex]d=\sqrt{(3-7)^{2}+(4-5)^{2}}[/tex]
[tex]d=\sqrt{(-4)^{2}+(-1)^{2}}[/tex]
[tex]DE=\sqrt{17}\ units[/tex]
step 2
Find the distance EF
[tex]E(4, 3),F(8, 2)[/tex]
substitute
[tex]d=\sqrt{(2-3)^{2}+(8-4)^{2}}[/tex]
[tex]d=\sqrt{(-1)^{2}+(4)^{2}}[/tex]
[tex]EF=\sqrt{17}\ units[/tex]
step 3
Find the distance DF
[tex]D(5, 7),F(8, 2)[/tex]
substitute
[tex]d=\sqrt{(2-7)^{2}+(8-5)^{2}}[/tex]
[tex]d=\sqrt{(-5)^{2}+(3)^{2}}[/tex]
[tex]DF=\sqrt{34}\ units[/tex]
step 4
The triangle DEF is a right triangle because satisfy the Pythagoras theorem
so
[tex](\sqrt{34})^{2}=(\sqrt{17})^{2}+(\sqrt{17})^{2}[/tex]
[tex]34=34[/tex] -----> is true
therefore
The measure of angle DEF is a right angle
