Find the lengths of the sides of the triangle by using the Distance Formula.
Classification: Scalene Is It a Right Triangle? No
Practice makes perfect
To begin, let's plot the given points on a coordinate plane and graph the triangle.
Now we will classify it as scalene, isosceles, or equilateral. Then we will determine whether it is a right triangle.
Classifying the Triangle
Let's begin by reviewing the definitions of a scalene, isosceles, and equilateral triangles.
Scalene: a triangle with three sides all of different lengths.
Isosceles: a triangle with two sides of equal length.
Equilateral: a triangle with all sides of equal length.
To classify our triangle, we will find the length of each side using the Distance Formula.
Side
Points
sqrt((x_2-x_1)^2+(y_2-y_1)^2)
Simplify
AB
A( 3,2), B( -10,4)
sqrt(( -10- 3)^2+( 4- 2)^2)
sqrt(173)
BC
B( -10,4), C( -5, -8)
sqrt(( -5-( -10))^2+( -8- 4)^2)
sqrt(169)
CA
C( -5, -8), A( 3,2)
sqrt(( 3-( -5))^2+( 2-( -8))^2)
sqrt(164)
As we can see, each side of our triangle has a different length. Therefore, it is a scalene triangle.
Is It a Right Triangle?
A right triangle is a triangle where one of the interior angles measures 90^(∘). This means that two sides are perpendicular. Let's find the slopes of the sides of our triangle and check whether any two are opposite reciprocals. We will do so using the Slope Formula.
Side
Points
y_2-y_1/x_2-x_1
Simplify
AB
A( 3,2), B( -10,4)
4- 2/-10- 3
- 2/13
BC
B( -10,4), C( -5, -8)
-8- 4/-5-( -10)
- 12/5
CA
C( -5, -8), A( 3,2)
2-( -8)/3-( -5)
5/4
We can tell that none of the pairs of slopes are opposite reciprocals.
- 2/13 (- 12/5 ) &≠ -1 [1em]
- 2/13 ( 5/4 ) &≠ -1 [1em]
- 12/5 ( 5/4 ) &≠ -1
Therefore, our triangle is not a right triangle.
