Question

what is true about △abc? choose three correct answers. a(-1,3) b(-5,-1) c(3,-1)
\\(\overline{ab}\perp\overline{ac}\\)
the triangle is an equilateral triangle.
\\(\overline{bc}\parallel\overline{ac}\\)
the triangle is an isosceles triangle.
the triangle is a right triangle.

Explanation:

Step1: Calculate the lengths of the sides

Use the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
For $AB$ with $A(-1,3)$ and $B(-5,-1)$:
$AB=\sqrt{(-1+ 5)^2+(3 + 1)^2}=\sqrt{4^2+4^2}=\sqrt{16 + 16}=\sqrt{32}=4\sqrt{2}$
For $AC$ with $A(-1,3)$ and $C(3,-1)$:
$AC=\sqrt{(-1 - 3)^2+(3 + 1)^2}=\sqrt{(-4)^2+4^2}=\sqrt{16 + 16}=\sqrt{32}=4\sqrt{2}$
For $BC$ with $B(-5,-1)$ and $C(3,-1)$:
$BC=\sqrt{(-5 - 3)^2+(-1+1)^2}=\sqrt{(-8)^2+0^2}=8$

Step2: Check the properties

Since $AB = AC=4\sqrt{2}$, the triangle is isosceles.
The slope of $AB$ is $m_{AB}=\frac{3 + 1}{-1+5}=\frac{4}{4} = 1$.
The slope of $AC$ is $m_{AC}=\frac{3 + 1}{-1 - 3}=\frac{4}{-4}=-1$.
Since $m_{AB}\times m_{AC}=1\times(-1)= - 1$, $\overline{AB}\perp\overline{AC}$, and the triangle is a right - triangle.

Answer:

$\overline{AB}\perp\overline{AC}$, The triangle is an isosceles triangle, The triangle is a right triangle.