Question

question 8 of 10
classify the triangle in the cartesian plane below by using the distance formula.
b(-2, 2)
a(4, 7)
c(7, 4)

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 $A(4,7)$ and $B(-2,2)$, we have $x_1 = 4,y_1 = 7,x_2=-2,y_2 = 2$. Then $AB=\sqrt{(-2 - 4)^2+(2 - 7)^2}=\sqrt{(-6)^2+(-5)^2}=\sqrt{36 + 25}=\sqrt{61}$.

Step3: Calculate length of BC

For points $B(-2,2)$ and $C(7,4)$, we have $x_1=-2,y_1 = 2,x_2 = 7,y_2=4$. Then $BC=\sqrt{(7+2)^2+(4 - 2)^2}=\sqrt{81+4}=\sqrt{85}$.

Step4: Calculate length of AC

For points $A(4,7)$ and $C(7,4)$, we have $x_1 = 4,y_1 = 7,x_2 = 7,y_2=4$. Then $AC=\sqrt{(7 - 4)^2+(4 - 7)^2}=\sqrt{9 + 9}=\sqrt{18}=3\sqrt{2}$.

Step5: Classify the triangle

Since all three side - lengths $\sqrt{61},\sqrt{85},3\sqrt{2}$ are different, the triangle is a scalene triangle.

Answer:

Scalene triangle