Question

use the triangles shown.
is \\( \overline { a c } \\) perpendicular to \\( \overline { x z } \\) ? explain.
a yes, the slope of \\( \overline { a c } \\) is 1, and the slope of \\( \overline { x z } \\) is -1.
b. yes, the slope of \\( \overline { a c } \\) is -1, and the slope of \\( \overline { x z } \\) is 1.
c. no, the slope of \\( \overline { a c } \\) is 1, and the slope of \\( \overline { x z } \\) is not -1, it is \\( \frac { 2 } { 3 } \\)
d. no, the slope of \\( \overline { a c } \\) is -1, and the slope of \\( \overline { x z } \\) is not 1, it is \\( \frac { 2 } { 3 } \\).

Explanation:

Step1: Calculate the slope of \(\overline{AC}\)

The formula for slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For points \(A(-1,3)\) and \(C(4,-2)\), \(m_{AC}=\frac{-2 - 3}{4-(-1)}=\frac{-5}{5}=-1\).

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

For points \(X(11,1)\) and \(Z(6,-2)\), \(m_{XZ}=\frac{1-(-2)}{11 - 6}=\frac{3}{5}=\frac{3}{5}
eq1\).
Two lines are perpendicular if \(m_1\times m_2=-1\). Here \(m_{AC}=-1\) and \(m_{XZ}=\frac{3}{5}\), and \(-1\times\frac{3}{5}
eq - 1\).

Answer:

D. No; the slope of \(\overline{AC}\) is \(-1\), and the slope of \(\overline{XZ}\) is not \(1\), it is \(\frac{3}{5}\)