the side lengths of triangle $rst$ are 5, 12, and 13. which set of ordered pairs form a triangle that is congruent to triangle $rst$?

$(-3,-2), (-3,11), (9,-2)$
$(1,-3), (6,-3), (6,10)$
$(2,-1), (7,-1), (2,11)$
$(5,1), (1,13), (12,1)$

Question

the side lengths of triangle $rst$ are 5, 12, and 13. which set of ordered pairs form a triangle that is congruent to triangle $rst$?

$(-3,-2), (-3,11), (9,-2)$
$(1,-3), (6,-3), (6,10)$
$(2,-1), (7,-1), (2,11)$
$(5,1), (1,13), (12,1)$

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 side lengths for the first option $$ \begin{aligned} &A(-3,-2), \quad B(-3,11), \quad C(9,-2) \ &d(A,B) = \sqrt{(-3 - (-3))^2 + (11 - (-2))^2} = \sqrt{0 + 13^2} = 13 \ &d(A,C) = \sqrt{(9 - (-3))^2 + (-2 - (-2))^2} = \sqrt{12^2 + 0} = 12 \ &d(B,C) = \sqrt{(9 - (-3))^2 + (-2 - 11)^2} = \sqrt{12^2 + (-13)^2} = \sqrt{144 + 169} = \sqrt{313} eq 5 \end{aligned} $$ ## Calculate side lengths for the second option $$ \begin{aligned} &A(1,-3), \quad B(6,-3), \quad C(6,10) \ &d(A,B) = \sqrt{(6-1)^2 + (-3 - (-3))^2} = \sqrt{5^2 + 0} = 5 \ &d(B,C) = \sqrt{(6-6)^2 + (10 - (-3))^2} = \sqrt{0 + 13^2} = 13 \ &d(A,C) = \sqrt{(6-1)^2 + (10 - (-3))^2} = \sqrt{5^2 + 13^2} = \sqrt{25 + 169} = \sqrt{194} eq 12 \end{aligned} $$ ## Calculate side lengths for the third option $$ \begin{aligned} &A(2,-1), \quad B(7,-1), \quad C(2,11) \ &d(A,B) = \sqrt{(7-2)^2 + (-1 - (-1))^2} = \sqrt{5^2 + 0} = 5 \ &d(A,C) = \sqrt{(2-2)^2 + (11 - (-1))^2} = \sqrt{0 + 12^2} = 12 \ &d(B,C) = \sqrt{(2-7)^2 + (11 - (-1))^2} = \sqrt{(-5)^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \end{aligned} $$

(-3,-2),(-3,11),(9,-2)

(1,-3),(6,-3),(6,10)

(2,-1),(7,-1),(2,11)

(5,1),(1,13),(12,1)

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

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Calculate side lengths for the first option

Calculate side lengths for the second option

Calculate side lengths for the third option

Answer: