Question

which statement proves that △xyz is an isosceles right triangle?
○ xz is not perpendicular to xy
○ xz≠xy
○ the slope of xz is 3/4, the slope of xy is -4/3, and xz = xy = 5.
○ the slope of xz is 3/4, the slope of xy is -4/3, and the slope of zy = 7.

Explanation:

Step1: Recall properties of isosceles right - triangle

An isosceles right - triangle has two equal sides and two perpendicular sides.

Step2: Check slope for perpendicularity

The product of slopes of two perpendicular lines is - 1. If the slope of $\overline{XZ}$ is $\frac{3}{4}$ and the slope of $\overline{XY}$ is $-\frac{4}{3}$, then $\frac{3}{4}\times(-\frac{4}{3})=- 1$, so $\overline{XZ}\perp\overline{XY}$.

Step3: Check side - length equality

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For $X(1,3)$ and $Z(5,6)$, $XZ=\sqrt{(5 - 1)^2+(6 - 3)^2}=\sqrt{16 + 9}=\sqrt{25}=5$. For $X(1,3)$ and $Y(4,-1)$, $XY=\sqrt{(4 - 1)^2+(-1 - 3)^2}=\sqrt{9 + 16}=\sqrt{25}=5$. So $XZ = XY$.

Answer:

The slope of $\overline{XZ}$ is $\frac{3}{4}$, the slope of $\overline{XY}$ is $-\frac{4}{3}$, and $XZ = XY = 5$.