which segment is the image of $\overline{ab}$ reflected across the line $y = -2$?

Question

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Coordinate Transformations", "Translation Rule" ], "new_concepts": [ "Reflection Across Horizontal Lines" ], "current_concepts": [ "Coordinate Transformations", "Translation Rule", "Reflection Across Horizontal Lines" ] } ## Identify coordinates of the original segment Using the **Coordinate Transformations** knowledge point $$ \begin{aligned} &A = (2, -7)\ &B = (8, -3) \end{aligned} $$ ## Apply reflection rule across a horizontal line Using the **Coordinate Transformations** and **Translation Rule** knowledge points $$ \begin{aligned} &(x, y) \to (x, 2k - y) \quad \text{for } y = k\ &k = -2\ &y' = 2(-2) - y = -4 - y \end{aligned} $$ ## Calculate the reflected coordinates Using the **Coordinate Transformations** knowledge point $$ \begin{aligned} &A'(x_A, y_A') = (2, -4 - (-7)) = (2, 3)\ &B'(x_B, y_B') = (8, -4 - (-3)) = (8, -1) \end{aligned} $$ ## Match with the segments on the graph Using the **Coordinate Transformations** knowledge point $$ \begin{aligned} &G = (2, 3)\ &H = (8, -1)\ &\text{Segment } GH \text{ connects } (2, 3) \text{ and } (8, -1) \end{aligned} $$ $\overline{GH}$ { "elements": [ { "type": "line", "params": [[-10, -2], [10, -2]], "properties": { "name": "y = -2", "strokeColor": "#F2557F", "strokeWidth": 2, "dash": 2, "withLabel": true, "label": { "position": "right" } } }, { "type": "segment", "params": [[2, -7], [8, -3]], "properties": { "name": "AB", "strokeColor": "#5583F2", "strokeWidth": 3, "withLabel": true } }, { "type": "segment", "params": [[2, 3], [8, -1]], "properties": { "name": "GH", "strokeColor": "#8C55F2", "strokeWidth": 3, "withLabel": true } }, { "type": "point", "params": [[2, -7]], "properties": { "name": "A", "color": "#5583F2" } }, { "type": "point", "params": [[8, -3]], "properties": { "name": "B", "color": "#5583F2" } }, { "type": "point", "params": [[2, 3]], "properties": { "name": "G", "color": "#8C55F2" } }, { "type": "point", "params": [[8, -1]], "properties": { "name": "H", "color": "#8C55F2" } } ] } { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Reflection Across Horizontal Lines" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Identify coordinates of the original segment

Apply reflection rule across a horizontal line

Calculate the reflected coordinates

Match with the segments on the graph

Answer: