Question

which statement proves that \\( \triangle xyz \\) is an isosceles right triangle?
\\( \bigcirc \\) \\( \overline{xz} \\) is not perpendicular to \\( \overline{xy} \\)
\\( \bigcirc \\) \\( xz \
eq xy \\)
\\( \bigcirc \\) the slope of \\( \overline{xz} \\) is \\( \frac{3}{4} \\), the slope of \\( \overline{xy} \\) is \\( -\frac{4}{3} \\), and \\( xz \\)
\\( =xy = 5 \\)
\\( \bigcirc \\) 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 sides of equal length (isosceles property) and the two legs are perpendicular (right - triangle property). The slope of a line passing through two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by \(m=\frac{y_2 - y_1}{x_2 - x_1}\), and two lines with slopes \(m_1\) and \(m_2\) are perpendicular if \(m_1\times m_2=- 1\). The distance between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).

Step2: Calculate the slope of \(XZ\)

For points \(X(1,3)\) and \(Z(5,6)\), the slope of \(\overline{XZ}\), \(m_{XZ}=\frac{6 - 3}{5 - 1}=\frac{3}{4}\).

Step3: Calculate the slope of \(XY\)

For points \(X(1,3)\) and \(Y(4,-1)\), the slope of \(\overline{XY}\), \(m_{XY}=\frac{-1 - 3}{4 - 1}=\frac{-4}{3}\).

Step4: Check the perpendicularity

Multiply the slopes: \(m_{XZ}\times m_{XY}=\frac{3}{4}\times(-\frac{4}{3})=-1\), so \(\overline{XZ}\perp\overline{XY}\).

Step5: Calculate the length of \(XZ\)

Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), for \(X(1,3)\) and \(Z(5,6)\), \(XZ=\sqrt{(5 - 1)^2+(6 - 3)^2}=\sqrt{16 + 9}=\sqrt{25}=5\).

Step6: Calculate the length of \(XY\)

For \(X(1,3)\) and \(Y(4,-1)\), \(XY=\sqrt{(4 - 1)^2+(-1 - 3)^2}=\sqrt{9 + 16}=\sqrt{25}=5\).

Since \(\overline{XZ}\perp\overline{XY}\) (right - angle) and \(XZ = XY\) (equal - length sides), the statement "The slope of \(\overline{XZ}\) is \(\frac{3}{4}\), the slope of \(\overline{XY}\) is \(-\frac{4}{3}\), and \(XZ=XY = 5\)" is correct.

Answer:

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