Question

classify △abc by the side lengths with the given points a(4, -2), b(5, 5), and c(-1, 3). △abc is a(n) select triangle.

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 $A(4,-2)$ and $B(5,5)$, we have $x_1 = 4,y_1=-2,x_2 = 5,y_2 = 5$. Then $AB=\sqrt{(5 - 4)^2+(5+ 2)^2}=\sqrt{1 + 49}=\sqrt{50}=5\sqrt{2}$.

Step3: Calculate length of $BC$

For points $B(5,5)$ and $C(-1,3)$, we have $x_1 = 5,y_1 = 5,x_2=-1,y_2 = 3$. Then $BC=\sqrt{(-1 - 5)^2+(3 - 5)^2}=\sqrt{(-6)^2+( - 2)^2}=\sqrt{36 + 4}=\sqrt{40}=2\sqrt{10}$.

Step4: Calculate length of $AC$

For points $A(4,-2)$ and $C(-1,3)$, we have $x_1 = 4,y_1=-2,x_2=-1,y_2 = 3$. Then $AC=\sqrt{(-1 - 4)^2+(3 + 2)^2}=\sqrt{(-5)^2+5^2}=\sqrt{25 + 25}=\sqrt{50}=5\sqrt{2}$.

Step5: Classify the triangle

Since $AB = AC=5\sqrt{2}$ and $BC = 2\sqrt{10}$, $\triangle ABC$ is an isosceles triangle.

Answer:

isosceles