Question

10 multiple choice 5 points
which statement proves that $\triangle xyz$ is an isosceles right triangle?
$xz \perp xy$
$xz = xy = 5$
the slope of xz is $\frac{3}{4}$, the slope of xy is $-\frac{4}{3}$, and $xz = xy = 5$.
the slope of xz is $\frac{3}{4}$, the slope of xy is $-\frac{4}{3}$, and the slope of zy = 7.

Explanation:

Step1: Calculate slope of XZ

Slope formula: $m=\frac{y_2-y_1}{x_2-x_1}$
For $X(1,3)$ and $Z(5,6)$:
$m_{XZ}=\frac{6-3}{5-1}=\frac{3}{4}$

Step2: Calculate slope of XY

For $X(1,3)$ and $Y(4,-1)$:
$m_{XY}=\frac{-1-3}{4-1}=\frac{-4}{3}$

Step3: Check perpendicularity

Two lines are perpendicular if $m_1 \times m_2 = -1$.
$m_{XZ} \times m_{XY} = \frac{3}{4} \times (-\frac{4}{3}) = -1$, so $XZ \perp XY$.

Step4: Calculate length of XZ

Distance formula: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
$XZ=\sqrt{(5-1)^2+(6-3)^2}=\sqrt{16+9}=\sqrt{25}=5$

Step5: Calculate length of XY

$XY=\sqrt{(4-1)^2+(-1-3)^2}=\sqrt{9+16}=\sqrt{25}=5$

Step6: Verify isosceles right triangle

We need both perpendicularity (right angle) and equal side lengths (isosceles).

Answer:

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