identifying a congruent triangle

which are the coordinates of the vertices of a triangle congruent to $\delta pqr$?

(0,5), (2,10), (0,10)
(9,-3), (9,0), (4,0)
(4,3), (-4,8), (-4,3)
(2.5,-4), (0,0), (0,-4)

Question

identifying a congruent triangle

which are the coordinates of the vertices of a triangle congruent to $\delta pqr$?

(0,5), (2,10), (0,10)
(9,-3), (9,0), (4,0)
(4,3), (-4,8), (-4,3)
(2.5,-4), (0,0), (0,-4)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Distance Formula", "Coordinate Geometry", "Triangle Congruence" ], "new_concepts": [], "current_concepts": [ "Distance Formula", "Coordinate Geometry", "Triangle Congruence" ] } ## Calculate the side lengths of $\Delta PQR$ $$ \begin{aligned} &P(0,0),\quad Q(0,-8),\quad R(5,-8)\ &PQ = \sqrt{(0-0)^2 + (-8-0)^2} = 8\ &QR = \sqrt{(5-0)^2 + (-8-(-8))^2} = 5\ &PR = \sqrt{(5-0)^2 + (-8-0)^2} = \sqrt{25 + 64} = \sqrt{89} \end{aligned} $$ ## Verify side lengths of the given options $$ \begin{aligned} &\text{Option 1: } A(0,5), B(2,10), C(0,10)\ &AC = 5,\quad BC = \sqrt{(0-2)^2 + (10-10)^2} = 2 e 8\ &\text{Option 2: } A(9,-3), B(9,0), C(4,0)\ &AB = 3 e 5 \text{ or } 8\ &\text{Option 3: } A(4,3), B(-4,8), C(-4,3)\ &AC = \sqrt{(-4-4)^2 + (3-3)^2} = 8\ &BC = \sqrt{(-4-(-4))^2 + (3-8)^2} = 5\ &AB = \sqrt{(-4-4)^2 + (8-3)^2} = \sqrt{64 + 25} = \sqrt{89} \end{aligned} $$ ## Confirm congruence by SSS $$ \begin{aligned} &\text{Side lengths of } \Delta PQR: \{5, 8, \sqrt{89}\}\ &\text{Side lengths of } \Delta ABC: \{5, 8, \sqrt{89}\}\ &\Delta PQR \cong \Delta ABC \text{ by SSS} \end{aligned} $$

(A) (0,5), (2,10), (0,10)

(B) (9,-3), (9,0), (4,0)

(C) (4,3), (-4,8), (-4,3)

(D) (2.5,-4), (0,0), (0,-4)

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

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QUESTION IMAGE

Question

Explanation:

Calculate the side lengths of \(\Delta PQR\)

Verify side lengths of the given options

Confirm congruence by SSS

Answer: