2. triangle $lmn$ with vertices $l(6, 6)$, $m(8, 8)$, and $n(8, 3)$:
a) reflection in the line $x = 5$
b) $270^\circ$ counterclockwise rotation about the origin

$l(\quad, \quad)$
$m(\quad, \quad)$
$n(\quad, \quad)$

Question

2. triangle $lmn$ with vertices $l(6, 6)$, $m(8, 8)$, and $n(8, 3)$:
a) reflection in the line $x = 5$
b) $270^\circ$ counterclockwise rotation about the origin

$l(\quad, \quad)$
$m(\quad, \quad)$
$n(\quad, \quad)$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Coordinate Reflection", "Sequence of Transformations" ], "new_concepts": [ "Coordinate Rotation", "Composite Transformations" ], "current_concepts": [ "Coordinate Reflection", "Coordinate Rotation", "Sequence of Transformations" ] } ## Identify the initial vertices We are given $\triangle LMN$ with vertices: $$L(6, 6), \quad M(8, 8), \quad N(8, 3)$$ ## Apply reflection in the line $x = 5$ Using the **Coordinate Reflection** knowledge point $$ \begin{aligned} &(x, y) \to (2h - x, y) \quad \text{where } h = 5\ &L'(2(5) - 6, 6) = L'(4, 6)\ &M'(2(5) - 8, 8) = M'(2, 8)\ &N'(2(5) - 8, 3) = N'(2, 3) \end{aligned} $$ ## Apply $270^\circ$ counterclockwise rotation about the origin Using the **Sequence of Transformations** knowledge point $$ \begin{aligned} &(x, y) \to (y, -x)\ &L''(6, -4)\ &M''(8, -2)\ &N''(3, -2) \end{aligned} $$ Triangle $LMN$ with vertices $L(6, 6)$, $M(8, 8)$, and $N(8, 3)$: a) reflection in the line $x = 5$ b) $270^\circ$ counterclockwise rotation about the origin $L'$ $(4, 6)$ $\to L''$ $(6, -4)$ $M'$ $(2, 8)$ $\to M''$ $(8, -2)$ $N'$ $(2, 3)$ $\to N''$ $(3, -2)$ { "elements": [ { "type": "polygon", "params": [[6, 6], [8, 8], [8, 3]], "properties": { "name": "LMN", "fillColor": "rgba(140, 85, 242, 0.1)", "strokeColor": "#8C55F2", "strokeWidth": 2, "withLabel": true } }, { "type": "line", "params": [[5, -10], [5, 10]], "properties": { "name": "x = 5", "strokeColor": "#F2557F", "strokeWidth": 1.5, "dash": 2, "withLabel": true } }, { "type": "polygon", "params": [[4, 6], [2, 8], [2, 3]], "properties": { "name": "L'M'N'", "fillColor": "rgba(85, 131, 242, 0.1)", "strokeColor": "#5583F2", "strokeWidth": 2, "dash": 1, "withLabel": true } }, { "type": "polygon", "params": [[6, -4], [8, -2], [3, -2]], "properties": { "name": "L''M''N''", "fillColor": "rgba(85, 221, 242, 0.15)", "strokeColor": "#55DDF2", "strokeWidth": 2, "withLabel": true } }, { "type": "point", "params": [[6, 6]], "properties": { "name": "L(6,6)", "color": "#8C55F2" } }, { "type": "point", "params": [[8, 8]], "properties": { "name": "M(8,8)", "color": "#8C55F2" } }, { "type": "point", "params": [[8, 3]], "properties": { "name": "N(8,3)", "color": "#8C55F2" } }, { "type": "point", "params": [[4, 6]], "properties": { "name": "L'(4,6)", "color": "#5583F2" } }, { "type": "point", "params": [[2, 8]], "properties": { "name": "M'(2,8)", "color": "#5583F2" } }, { "type": "point", "params": [[2, 3]], "properties": { "name": "N'(2,3)", "color": "#5583F2" } }, { "type": "point", "params": [[6, -4]], "properties": { "name": "L''(6,-4)", "color": "#55DDF2" } }, { "type": "point", "params": [[8, -2]], "properties": { "name": "M''(8,-2)", "color": "#55DDF2" } }, { "type": "point", "params": [[3, -2]], "properties": { "name": "N''(3,-2)", "color": "#55DDF2" } } ] } { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Coordinate Rotation" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the initial vertices

Apply reflection in the line \(x = 5\)

Apply \(270^\circ\) counterclockwise rotation about the origin

Answer: