[tex]D(7,3) \\
E(8,1) \\
F(4,-1)[/tex]
The lengths of the sides of the triangle:
[tex]\overline{DE}=\sqrt{(8-7)^2+(1-3)^2}=\sqrt{1^2+(-2)^2}=\sqrt{1+4}=\sqrt{5} \\ \\
\overline{DF}=\sqrt{(4-7)^2+(-1-3)^2}=\sqrt{(-3)^2+(-4)^2}=\sqrt{9+16}=\sqrt{25}=5 \\ \\
\overline{EF}=\sqrt{(4-8)^2+(-1-1)^2}=\sqrt{(-4)^2+(-2)^2}=\sqrt{16+4}=\sqrt{20}= \\
=\sqrt{4 \times 5}=2\sqrt{5}[/tex]
No sides are equal to each other, so it's a scalene triangle.
Check if the sides satisfy the Pythagorean theorem:
[tex](\sqrt{5})^2 + (2\sqrt{5})^2=5^2 \\
5+20=25 \\
25=25 \\
true[/tex]
They do, so it's a right triangle.
The triangle is a right scalene triangle.
