lydia graphed $\delta xyz$ at the coordinates $x(0, -4)$, $y(2, -3)$, and $z(2, -6)$. she thinks $\delta xyz$ is a right triangle. is lydias assertion correct?

yes, the slopes of $\overline{xy}$ and $\overline{xz}$ are the same.
yes, the slopes of $\overline{xy}$ and $\overline{xz}$ are opposite reciprocals.
no, the slopes of $\overline{xy}$ and $\overline{xz}$ are not the same.
no, the slopes of $\overline{xy}$ and $\overline{xz}$ are not opposite reciprocals.

Question

lydia graphed $\delta xyz$ at the coordinates $x(0, -4)$, $y(2, -3)$, and $z(2, -6)$. she thinks $\delta xyz$ is a right triangle. is lydias assertion correct?

yes, the slopes of $\overline{xy}$ and $\overline{xz}$ are the same.
yes, the slopes of $\overline{xy}$ and $\overline{xz}$ are opposite reciprocals.
no, the slopes of $\overline{xy}$ and $\overline{xz}$ are not the same.
no, the slopes of $\overline{xy}$ and $\overline{xz}$ are not opposite reciprocals.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Perpendicular Lines", "Coordinate Geometry Proofs" ], "new_concepts": [], "current_concepts": [ "Perpendicular Lines", "Coordinate Geometry Proofs", "Right Triangle Properties" ] } ## Calculate the slopes of the segments $$ \begin{aligned} m_{\overline{XY}} &= \frac{-3 - (-4)}{2 - 0} = \frac{1}{2} \ m_{\overline{XZ}} &= \frac{-6 - (-4)}{2 - 0} = \frac{-2}{2} = -1 \ m_{\overline{YZ}} &= \text{undefined (vertical line since } x_Y = x_Z = 2) \end{aligned} $$ ## Check for perpendicularity $$ \begin{aligned} m_{\overline{XY}} \cdot m_{\overline{XZ}} &= \frac{1}{2} \cdot (-1) = -\frac{1}{2} eq -1 \ \text{Slopes of } \overline{XY} \text{ and } \overline{XZ} &\text{ are not opposite reciprocals.} \end{aligned} $$ ## Determine the correct option $$ \text{Since the slopes of } \overline{XY} \text{ and } \overline{XZ} \text{ are not opposite reciprocals, they are not perpendicular, so } \Delta XYZ \text{ is not a right triangle.} $$

Yes, the slopes of $\overline{XY}$ and $\overline{XZ}$ are the same.

Yes, the slopes of $\overline{XY}$ and $\overline{XZ}$ are opposite reciprocals.

No, the slopes of $\overline{XY}$ and $\overline{XZ}$ are not the same.

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

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QUESTION IMAGE

Question

Explanation:

Calculate the slopes of the segments

Check for perpendicularity

Determine the correct option

Answer: