Find the lengths of the sides of the triangle by using the Distance Formula.
Classification: Isosceles. Is It a Right Triangle? Yes.
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 is a triangle with three sides all of different lengths.
Isoscelesis a triangle with two sides of equal length.
Equilateral is 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
HB
H( 1,- 2), B( - 1,4)
sqrt(( - 1- 1)^2+( 4-( - 2))^2)
sqrt(40)
BF
B( - 1,4), F( 5, 6)
sqrt(( 5-( - 1))^2+( 6- 4)^2)
sqrt(40)
FH
F( 5, 6), H( 1,- 2)
sqrt(( 1- 5)^2+( - 2- 6)^2)
sqrt(80)
As we can see, the triangle has two sides of equal length. Therefore, it is an isosceles 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 this using the Slope Formula.
Side
Points
y_2-y_1/x_2-x_1
Simplify
HB
H( 1,- 2), B( - 1,4)
4-( - 2)/- 1- 1
- 3
BF
B( - 1,4), F( 5, 6)
6- 4/5-( - 1)
1/3
FH
F( 5, 6), H( 1,- 2)
- 2- 6/1- 5
2
Notice that - 3 and 13 are opposite reciprocals.
- 3 * 13= -1
Therefore, our triangle is a right triangle.
