Question

A triangle with vertices (3,-2) (7,3) and (-7,6) is what type of triangle ?

A) equilateral
B) obtuse scalene
C)isosceles right
D) right

Answer 1

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):

`d=√((-7-3)^2+(6--2)^2) \\ \\=√((-10)^2+8^2) = √(100+64)=√(164)`

The side from (-7, 6) to (7, 3) (using the distance formula:

`d=√((-7-7)^2+(6-3)^2) \\ \\=√((-14)^2+3^2)=√(196+9)=√(205)`

The sides are different, so the triangle is only right.

Answer 2

D) right
The slope between (3,-2) (7,3) and (3,-2) (-7,6) are opposite reciprocals of 5/4 and -4/5. If the slopes are opposite reciprocals they are perpendicular and have an angle equal to 90 degrees which makes it a right triangle. none of the distances are equal so it cannot be an isosceles right triangle.