To classify the triangle by its sides, you need to know their lengths. To determine if the triangle is a right triangle, you need to know the slopes of the sides.
Classify by sides: Isosceles triangle Right triangle? Yes
Now we can classify the triangle by calculating the side lengths and then we can determine if it is a right triangle.
Classify by Sides
To classify a triangle by its sides we need to determine if it is scalene, isosceles, or equilateral. To do that, we have to calculate the length of the sides using the Distance Formula. Let's start by finding the distance between A( 0, 0) and B( 3, 3). This will give us the value of AB.
We can find the lengths of the other sides in the same way.
Side
Points
sqrt((x_2-x_1)^2+(y_2-y_1)^2)
Length
AB
( 0,0) & ( 3,3)
sqrt(( 3- 0)^2+( 3- 0)^2)
3sqrt(2)
AC
( 0,0) & ( - 3,3)
sqrt(( - 3- 0)^2+( 3- 0)^2)
3sqrt(2)
BC
( 3,3) & ( - 3,3)
sqrt(( - 3- 3)^2+( 3- 3)^2)
6
As we can see, AB and AC have the same length. Since two of the sides are congruent, the triangle is isosceles.
Right Triangle?
In our diagram, we see that ∠ B and ∠ C are acute angles. Therefore, if △ ABC is a right triangle, the right angle must be ∠ A. To determine if this is the case, we will first calculate the slope of CA and AB using the Slope Formula.
Side
Points
y_2-y_1/x_2-x_1
Slope
Simplified Slope
CA
( - 3,3) & ( 0,0)
0- 3/0-( - 3)
- 3/3
- 1
AB
( 0,0) & ( 3,3)
3- 0/3- 0
3/3
1
Since 1 and - 1 are opposite reciprocals, we know that AC is perpendicular to AB. Therefore, △ ABC is in fact a right triangle.
