Question

which statement proves that \\( \triangle 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: Recall the properties of an isosceles right - triangle

An isosceles right - triangle has two equal - length sides and the product of the slopes of the two perpendicular sides is \(- 1\). The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), and the slope formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).

Step2: Calculate the slope of \(\overline{XZ}\)

For points \(X(1,3)\) and \(Z(5,6)\), using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), we have \(m_{XZ}=\frac{6 - 3}{5 - 1}=\frac{3}{4}\).

Step3: Calculate the slope of \(\overline{XY}\)

For points \(X(1,3)\) and \(Y(4,-1)\), using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), we have \(m_{XY}=\frac{-1 - 3}{4 - 1}=\frac{-4}{3}\). The product of the slopes \(m_{XZ}\times m_{XY}=\frac{3}{4}\times(-\frac{4}{3})=-1\), so \(\overline{XZ}\perp\overline{XY}\).

Step4: Calculate the length of \(XZ\) and \(XY\)

Using the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)
For \(XZ\): \(x_1 = 1,y_1 = 3,x_2 = 5,y_2 = 6\), \(XZ=\sqrt{(5 - 1)^2+(6 - 3)^2}=\sqrt{16 + 9}=\sqrt{25}=5\)
For \(XY\): \(x_1 = 1,y_1 = 3,x_2 = 4,y_2=-1\), \(XY=\sqrt{(4 - 1)^2+(-1 - 3)^2}=\sqrt{9 + 16}=\sqrt{25}=5\)

Answer:

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