Question

classify the triangle in the cartesian plane below by using the distance formula.
a(-3,3)
b(-1,-4)
c(3,1)

Explanation:

Step1: Recall distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

Step2: Calculate length of AB

For points $A(-3,3)$ and $B(-1,-4)$, we have $x_1=-3,y_1 = 3,x_2=-1,y_2=-4$.
$AB=\sqrt{(-1-(-3))^2+(-4 - 3)^2}=\sqrt{(2)^2+(-7)^2}=\sqrt{4 + 49}=\sqrt{53}$.

Step3: Calculate length of BC

For points $B(-1,-4)$ and $C(3,1)$, we have $x_1=-1,y_1=-4,x_2=3,y_2 = 1$.
$BC=\sqrt{(3-(-1))^2+(1-(-4))^2}=\sqrt{(4)^2+(5)^2}=\sqrt{16 + 25}=\sqrt{41}$.

Step4: Calculate length of AC

For points $A(-3,3)$ and $C(3,1)$, we have $x_1=-3,y_1 = 3,x_2=3,y_2 = 1$.
$AC=\sqrt{(3-(-3))^2+(1 - 3)^2}=\sqrt{(6)^2+(-2)^2}=\sqrt{36+4}=\sqrt{40}=2\sqrt{10}$.
Since all three side - lengths $\sqrt{53},\sqrt{41},2\sqrt{10}$ are different, the triangle is a scalene triangle.

Answer:

The triangle is a scalene triangle.