triangle $def$ is congruent to right triangle $abc$, shown below. if point $f$ has the coordinates $(3, 4)$, what could be the coordinates of point $e$?

$(4, -1)$
$(3, -2)$
$(-2, 4)$
$(4, -2)$

Question

triangle $def$ is congruent to right triangle $abc$, shown below. if point $f$ has the coordinates $(3, 4)$, what could be the coordinates of point $e$?

$(4, -1)$
$(3, -2)$
$(-2, 4)$
$(4, -2)$

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" ] } ## Determine the coordinates of the vertices of triangle ABC $$ A = (-2, 0),\quad B = (-5, 2),\quad C = (-4, -3) $$ ## Calculate the squared side lengths of triangle ABC $$ \begin{aligned} AB^2 &= (-5 - (-2))^2 + (2 - 0)^2 = (-3)^2 + 2^2 = 9 + 4 = 13 \ BC^2 &= (-4 - (-5))^2 + (-3 - 2)^2 = 1^2 + (-5)^2 = 1 + 25 = 26 \ AC^2 &= (-4 - (-2))^2 + (-3 - 0)^2 = (-2)^2 + (-3)^2 = 4 + 9 = 13 \end{aligned} $$ ## Use congruence to find the coordinates of E $$ \begin{aligned} &\triangle DEF \cong \triangle ABC \implies DE^2 = AB^2 = 13,\quad EF^2 = BC^2 = 26,\quad DF^2 = AC^2 = 13 \ &D = (1, 1),\quad F = (3, 4) \ &\text{Test option } E = (4, -2): \ &DE^2 = (4 - 1)^2 + (-2 - 1)^2 = 3^2 + (-3)^2 = 9 + 9 = 18 eq 13 \ &\text{Test option } E = (3, -2): \ &DE^2 = (3 - 1)^2 + (-2 - 1)^2 = 2^2 + (-3)^2 = 4 + 9 = 13 \ &EF^2 = (3 - 3)^2 + (-2 - 4)^2 = 0^2 + (-6)^2 = 36 eq 26 \ &\text{Test option } E = (4, -1): \ &DE^2 = (4 - 1)^2 + (-1 - 1)^2 = 3^2 + (-2)^2 = 9 + 4 = 13 \ &EF^2 = (4 - 3)^2 + (-1 - 4)^2 = 1^2 + (-5)^2 = 1 + 25 = 26 \end{aligned} $$

(A) (4,-1)

(B) (3,-2)

(C) (-2,4)

(D) (4,-2)

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

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QUESTION IMAGE

Question

Explanation:

Determine the coordinates of the vertices of triangle ABC

Calculate the squared side lengths of triangle ABC

Use congruence to find the coordinates of E

Answer: