what is the area of triangle abc?

3 square units
7 square units
11 square units
15 square units

Question

what is the area of triangle abc?

3 square units
7 square units
11 square units
15 square units

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Triangle Area in Coordinate Plane", "Composite Area" ], "new_concepts": [], "current_concepts": [ "Triangle Area in Coordinate Plane", "Composite Area" ] } ## Identify the coordinates of the vertices Using the *Triangle Area in Coordinate Plane* knowledge point $$ A = (1, 1), \quad B = (4, 0), \quad C = (3, 5) $$ ## Calculate the area using bounding box subtraction Using the *Composite Area* knowledge point $$ \begin{aligned} &\text{Bounding Box: } [1, 4] \times [0, 5]\ &\text{Width } = 3, \quad \text{Height } = 5\ &\text{Box Area } = 3 \times 5 = 15\ &\text{Left Triangle Area } = \frac{1}{2} \times (3 - 1) \times (5 - 1) = 4\ &\text{Right Triangle Area } = \frac{1}{2} \times (4 - 3) \times (5 - 0) = 2.5\ &\text{Bottom Triangle Area } = \frac{1}{2} \times (4 - 1) \times (1 - 0) = 1.5\ &\text{Triangle Area } = 15 - (4 + 2.5 + 1.5) = 7 \end{aligned} $$

3 square units

7 square units

11 square units

15 square units

{ "elements": [ { "type": "point", "params": [[1, 1]], "properties": { "name": "A", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[4, 0]], "properties": { "name": "B", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[3, 5]], "properties": { "name": "C", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "polygon", "params": [[1, 1], [4, 0], [3, 5]], "properties": { "fillColor": "rgba(140, 85, 242, 0.3)", "strokeColor": "#8C55F2", "strokeWidth": 2 } }, { "type": "segment", "params": [[1, 0], [1, 5]], "properties": { "strokeColor": "#F2557F", "strokeWidth": 1, "dash": 2 } }, { "type": "segment", "params": [[4, 0], [4, 5]], "properties": { "strokeColor": "#F2557F", "strokeWidth": 1, "dash": 2 } }, { "type": "segment", "params": [[1, 5], [4, 5]], "properties": { "strokeColor": "#F2557F", "strokeWidth": 1, "dash": 2 } }, { "type": "segment", "params": [[1, 0], [4, 0]], "properties": { "strokeColor": "#F2557F", "strokeWidth": 1, "dash": 2 } } ] } { "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Triangle Area in Coordinate Plane" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the coordinates of the vertices

Calculate the area using bounding box subtraction

Answer: