It is a right triangle.
We can tell if it is right by finding the slope of the line segments making up the triangle. Two segments that are perpendicular have slopes that are negative reciprocals (opposite signs and flipped upside down).
The slope from (3, -2) to (7, 3) is
m = (-2-3)/(3-7) = -5/4.
The slope from (3, -2) to (-7, 6) is
m = (-2-6)/(3--7) = -8/10 = -4/5
These have opposite signs and are flipped, so they make a right angle.
We now check the sides. After plotting the points, we can see that two of the sides are noticeably longer than the third, so we will find their lengths.
The side from (-7, 6) to (3, -2) (using the distance formula):
[tex]d=\sqrt{(-7-3)^2+(6--2)^2}
\\
\\=\sqrt{(-10)^2+8^2} = \sqrt{100+64}=\sqrt{164}[/tex]
The side from (-7, 6) to (7, 3) (using the distance formula:
[tex]d=\sqrt{(-7-7)^2+(6-3)^2}
\\
\\=\sqrt{(-14)^2+3^2}=\sqrt{196+9}=\sqrt{205}[/tex]
The sides are different, so the triangle is only right.
