Question

for the given triangle, the slope of \\( \overline { z x } \\) is \\( \frac { 1 } { 3 } \\), the slope of \\( \overline { z y } \\) is \\( \frac { - 1 } { 2 } \\), and the slope of \\( \overline { x y } \\) is 2. which statement verifies that triangle xyz is a right triangle? the slopes of \\( \overline { z x } \\) and \\( \overline { z y } \\) are opposite reciprocals. the slopes of \\( \overline { x y } \\) and \\( \overline { z y } \\) are opposite reciprocals. the slopes of \\( \overline { z x } \\) and \\( \overline { z y } \\) have opposite signs. the slopes of \\( \overline { x y } \\) and \\( \overline { z y } \\) have opposite signs.

Explanation:

To determine if a triangle is a right triangle, we check if two sides are perpendicular. Two lines are perpendicular if their slopes are opposite reciprocals (product is -1).

• Slope of \( \overline{XY} \) is 2, slope of \( \overline{ZY} \) is \( \frac{-1}{2} \).
• Multiply them: \( 2\times\frac{-1}{2} = -1 \), so they are opposite reciprocals, meaning \( \overline{XY} \perp \overline{ZY} \), confirming a right angle.
• Other options: Slopes of \( \overline{ZX} (\frac{1}{3}) \) and \( \overline{ZY} (\frac{-1}{2}) \) multiply to \( \frac{-1}{6}

eq -1 \), so not opposite reciprocals. Opposite signs alone don't imply perpendicularity.

Answer:

The slopes of \( \overline{XY} \) and \( \overline{ZY} \) are opposite reciprocals.