Question

which statement proves that △xyz is an isosceles right triangle?
(overline{xz}) is not perpendicular to (overline{xy})
(xz
eq xy)
the slope of (overline{xz}) is (\frac{3}{4}), the slope of (overline{xy}) is (-\frac{4}{3}), and (xz = xy = 5).
the slope of (overline{xz}) is (\frac{3}{4}), the slope of (overline{xy}) is (-\frac{4}{3}), and the slope of (overline{zy}=7).

Explanation:

Step1: Calculate the length of \(XZ\) and \(XY\) using the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

For \(X(1,3)\) and \(Z(5,6)\):

$$
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$$

For \(X(1,3)\) and \(Y(4,-1)\):

$$
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$$

Step2: Check the slopes of \(XZ\) and \(XY\) using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\)

Slope of \(XZ\) with \(X(1,3)\) and \(Z(5,6)\):

$$m_{XZ}=\frac{6 - 3}{5 - 1}=\frac{3}{4}$$

Slope of \(XY\) with \(X(1,3)\) and \(Y(4,-1)\):

$$m_{XY}=\frac{-1 - 3}{4 - 1}=\frac{-4}{3}$$

Since \(m_{XZ}\times m_{XY}=\frac{3}{4}\times(-\frac{4}{3})=- 1\), \(XZ\perp XY\)

Answer:

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