which rule describes the composition of transformations that maps $\delta jkl$ to $\delta jkl$?

$r_{o,\, 90^\circ} \circ t_{0,\, -2}(x, y)$
$t_{0,\, -2} \circ r_{o,\, 90^\circ}(x, y)$
$r_{o,\, 90^\circ} \circ t_{-2,\, 0}(x, y)$
$t_{-2,\, 0} \circ r_{o,\, 90^\circ}(x, y)$

Question

which rule describes the composition of transformations that maps $\delta jkl$ to $\delta jkl$?

$r_{o,\, 90^\circ} \circ t_{0,\, -2}(x, y)$
$t_{0,\, -2} \circ r_{o,\, 90^\circ}(x, y)$
$r_{o,\, 90^\circ} \circ t_{-2,\, 0}(x, y)$
$t_{-2,\, 0} \circ r_{o,\, 90^\circ}(x, y)$

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" ], "new_concepts": [], "current_concepts": [ "Composition of Transformations", "Geometric Rotations", "Geometric Translations" ] } ## Identify coordinates of the original and final triangles Find the coordinates of the vertices of $\Delta JKL$ and $\Delta J''K''L''$ from the grid. $$ \begin{aligned} &J = (2, -4), \quad K = (5, -4), \quad L = (2, -2)\ &J'' = (2, 2), \quad K'' = (2, 5), \quad L'' = (0, 2) \end{aligned} $$ ## Analyze the transformation sequence Using the **Composition of Transformations** and **Geometric Rotations** knowledge points $$ \begin{aligned} &\text{Let } (x, y) \text{ be the original coordinates.}\ &\text{Apply rotation } R_{O, 90^\circ}(x, y) = (-y, x):\ &J(2, -4) \to J'(4, 2)\ &K(5, -4) \to K'(4, 5)\ &L(2, -2) \to L'(2, 2)\ &\text{Apply translation } T_{-2, 0}(x', y') = (x' - 2, y'):\ &J'(4, 2) \to J''(2, 2)\ &K'(4, 5) \to K''(2, 5)\ &L'(2, 2) \to L''(0, 2) \end{aligned} $$ ## Write the composition rule Using the **Composition of Transformations** knowledge point $$ \begin{aligned} &\text{The sequence is rotation first, then translation.}\ &\text{In function notation: } (T_{-2, 0} \circ R_{O, 90^\circ})(x, y) \end{aligned} $$

(A) $R_{O, 90^\circ} \circ T_{0, -2}(x, y)$

(B) $T_{0, -2} \circ R_{O, 90^\circ}(x, y)$

(C) $R_{O, 90^\circ} \circ T_{-2, 0}(x, y)$

(D) $T_{-2, 0} \circ R_{O, 90^\circ}(x, y)$

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

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QUESTION IMAGE

Question

Explanation:

Identify coordinates of the original and final triangles

Analyze the transformation sequence

Write the composition rule

Answer: