what is true about $\delta abc$? select three options

$\overline{ab} \perp \overline{ac}$

the triangle is a right triangle.

the triangle is an isosceles triangle.

the triangle is an equilateral triangle.

$\overline{bc} \parallel \overline{ac}$

Question

what is true about $\delta abc$? select three options

$\overline{ab} \perp \overline{ac}$

the triangle is a right triangle.

the triangle is an isosceles triangle.

the triangle is an equilateral triangle.

$\overline{bc} \parallel \overline{ac}$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Coordinate Geometry", "Distance Formula", "Perpendicular Slopes" ], "new_concepts": [], "current_concepts": [ "Coordinate Geometry", "Distance Formula", "Perpendicular Slopes" ] } ## Calculate slopes of the triangle sides $$ \begin{aligned} &A(-1, 3),\quad B(-5, -1),\quad C(3, -1)\ &\text{Slope of } \overline{AB} = \frac{-1 - 3}{-5 - (-1)} = \frac{-4}{-4} = 1\ &\text{Slope of } \overline{AC} = \frac{-1 - 3}{3 - (-1)} = \frac{-4}{4} = -1\ &\text{Slope of } \overline{BC} = \frac{-1 - (-1)}{3 - (-5)} = 0 \end{aligned} $$ ## Verify perpendicularity and right triangle properties $$ \begin{aligned} &(\text{Slope of } \overline{AB}) \cdot (\text{Slope of } \overline{AC}) = 1 \cdot (-1) = -1 \implies \overline{AB} \perp \overline{AC}\ &\angle A = 90^\circ \implies \triangle ABC \text{ is a right triangle} \end{aligned} $$ ## Calculate side lengths to determine triangle type $$ \begin{aligned} &AB = \sqrt{(-5 - (-1))^2 + (-1 - 3)^2} = \sqrt{(-4)^2 + (-4)^2} = \sqrt{32} = 4\sqrt{2}\ &AC = \sqrt{(3 - (-1))^2 + (-1 - 3)^2} = \sqrt{4^2 + (-4)^2} = \sqrt{32} = 4\sqrt{2}\ &BC = \sqrt{(3 - (-5))^2 + (-1 - (-1))^2} = \sqrt{8^2 + 0^2} = 8\ &AB = AC eq BC \implies \triangle ABC \text{ is an isosceles triangle} \end{aligned} $$

$\overline{AB} \perp \overline{AC}$

The triangle is a right triangle.

The triangle is an isosceles triangle.

The triangle is an equilateral triangle.

$\overline{BC} \parallel \overline{AC}$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Coordinate Geometry" ] }

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Question

Explanation:

Calculate slopes of the triangle sides

Verify perpendicularity and right triangle properties

Calculate side lengths to determine triangle type

Answer: