what are possible coordinates of point $d$ if $\delta def$ is congruent to $\delta abc$?

$(-6,1)$
$(-6,2)$
$(-5,1)$
$(-5,2)$

Question

what are possible coordinates of point $d$ if $\delta def$ is congruent to $\delta abc$?

$(-6,1)$
$(-6,2)$
$(-5,1)$
$(-5,2)$

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" ] } ## Identify coordinates of the given vertices $$ \begin{aligned} A &= (1, 0) \ B &= (-2, -5) \ C &= (4, -5) \ E &= (-1, -1) \ F &= (-1, 5) \end{aligned} $$ ## Calculate side lengths of triangle ABC $$ \begin{aligned} BC &= 4 - (-2) = 6 \ AB &= \sqrt{(1 - (-2))^2 + (0 - (-5))^2} = \sqrt{3^2 + 5^2} = \sqrt{34} \ AC &= \sqrt{(1 - 4)^2 + (0 - (-5))^2} = \sqrt{(-3)^2 + 5^2} = \sqrt{34} \end{aligned} $$ ## Determine coordinates of point D for congruence $$ \begin{aligned} &\Delta DEF \cong \Delta ABC \implies EF = BC = 6 \ &EF = \sqrt{(-1 - (-1))^2 + (5 - (-1))^2} = \sqrt{0^2 + 6^2} = 6 \ &DE = AB = \sqrt{34} \implies (x_D - (-1))^2 + (y_D - (-1))^2 = 34 \ &DF = AC = \sqrt{34} \implies (x_D - (-1))^2 + (y_D - 5)^2 = 34 \ &(y_D + 1)^2 = (y_D - 5)^2 \implies y_D = 2 \ &(x_D + 1)^2 + (2 + 1)^2 = 34 \implies (x_D + 1)^2 = 25 \implies x_D + 1 = \pm 5 \ &x_D = 4 \quad \text{or} \quad x_D = -6 \ &D = (4, 2) \quad \text{or} \quad D = (-6, 2) \end{aligned} $$

(A) $(-6,1)$

(B) $(-6,2)$

(C) $(-5,1)$

(D) $(-5,2)$

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

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QUESTION IMAGE

Question

Explanation:

Identify coordinates of the given vertices

Calculate side lengths of triangle ABC

Determine coordinates of point D for congruence

Answer: