square a\b\c\d\ is the final image after the rule $t_{-4, -1} \circ r_{0, 90^{\circ}}(x, y)$ was applied to square abcd.

what are the coordinates of vertex a of square abcd?

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

Question

square a\b\c\d\ is the final image after the rule $t_{-4, -1} \circ r_{0, 90^{\circ}}(x, y)$ was applied to square abcd.

what are the coordinates of vertex a of square abcd?

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

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Composition of Transformations", "Geometric Rotations", "Geometric Translations", "Inverse Transformations" ], "new_concepts": [], "current_concepts": [ "Composition of Transformations", "Geometric Rotations", "Geometric Translations", "Inverse Transformations" ] } ## Identify the coordinates of vertex A'' Using the *Composition of Transformations* knowledge point $$ A'' = (-4, -3) $$ ## Set up the transformation equation Using the *Composition of Transformations* and *Geometric Translations* knowledge points $$ (T_{-4, -1} \circ R_{O, 90^\circ})(A) = A'' $$ ## Apply the inverse translation Using the *Geometric Translations* and *Inverse Transformations* knowledge points $$ \begin{aligned} T_{-4, -1}(x', y') &= (x' - 4, y' - 1) = (-4, -3) \ x' - 4 &= -4 \implies x' = 0 \ y' - 1 &= -3 \implies y' = -2 \ A' &= (0, -2) \end{aligned} $$ ## Apply the inverse rotation Using the *Geometric Rotations* and *Inverse Transformations* knowledge points $$ \begin{aligned} R_{O, 90^\circ}(x, y) &= (-y, x) = (0, -2) \ -y &= 0 \implies y = 0 \ x &= -2 \ A &= (-2, 0) \end{aligned} $$ ## Re-evaluate the coordinates of A'' from the graph Using the *Composition of Transformations* knowledge point $$ \begin{aligned} A'' &= (-5, -3) \ T_{-4, -1}(x', y') &= (x' - 4, y' - 1) = (-5, -3) \ x' - 4 &= -5 \implies x' = -1 \ y' - 1 &= -3 \implies y' = -2 \ A' &= (-1, -2) \ R_{O, 90^\circ}(x, y) &= (-y, x) = (-1, -2) \ -y &= -1 \implies y = 1 \ x &= -2 \ A &= (-2, 1) \end{aligned} $$

(A) $(-1, -6)$

(B) $(-1, -2)$

(C) $(-1, 6)$

(D) $(-2, 1)$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Composition of Transformations" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the coordinates of vertex A''

Set up the transformation equation

Apply the inverse translation

Apply the inverse rotation

Re-evaluate the coordinates of A'' from the graph

Answer: