Question

question 3 (multiple choice worth 1 points)
(04.01r mc)
lydia graphed \\( \triangle x y z \\) at the coordinates \\( x(0,-4), y(2,-3) \\), and \\( z(2,-6) \\). she thinks \\( \triangle x y z \\) is a right triangle. is lydias assertion correct?
yes, the slopes of \\( \overline{x y} \\) and \\( \overline{x z} \\) are the same.
yes, the slopes of \\( \overline{x y} \\) and \\( \overline{x z} \\) are opposite reciprocals.
no, the slopes of \\( \overline{x y} \\) and \\( \overline{x z} \\) are not the same.
no, the slopes of \\( \overline{x y} \\) and \\( \overline{x z} \\) are not opposite reciprocals.

Explanation:

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

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For points \(X(0,-4)\) and \(Y(2,-3)\), \(m_{XY}=\frac{-3-(-4)}{2 - 0}=\frac{-3 + 4}{2}=\frac{1}{2}\)

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

For points \(X(0,-4)\) and \(Z(2,-6)\), \(m_{XZ}=\frac{-6-(-4)}{2-0}=\frac{-6 + 4}{2}=\frac{-2}{2}=-1\)

Step3: Check the relationship between the slopes

The product of the slopes \(m_{XY}\times m_{XZ}=\frac{1}{2}\times(-1)=-\frac{1}{2}
eq - 1\). Also, \(m_{XY}
eq m_{XZ}\)

Answer:

No, the slopes of \(\overline{XY}\) and \(\overline{XZ}\) are not opposite reciprocals.