verifying a triangle is a right triangle
for the given triangle, the slope of $ \overline{zx} $ is $ \frac{1}{3} $, the slope of
$ \overline{zy} $ is $ \frac{-1}{2} $, and the slope of $ \overline{xy} $ is 2. which statement
verifies that triangle xyz is a right triangle?
the slopes of $ \overline{zx} $ and $ \overline{zy} $ are opposite reciprocals.
the slopes of $ \overline{xy} $ and $ \overline{zy} $ are opposite reciprocals.
the slopes of $ \overline{zx} $ and $ \overline{zy} $ have opposite signs.
the slopes of $ \overline{xy} $ and $ \overline{zy} $ have opposite signs.

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# Brief Explanations:
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} $.
- Product of $ 2 $ and $ \frac{-1}{2} $ is $ 2\times\frac{-1}{2}=-1 $, so they are opposite reciprocals, meaning $ \overline{XY} $ and $ \overline{ZY} $ are perpendicular.
- Other options: Slopes of $ \overline{ZX} $ ($ \frac{1}{3} $) and $ \overline{ZY} $ ($ \frac{-1}{2} $): product $ \frac{1}{3}\times\frac{-1}{2}=\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.

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