type the correct answer in each box. if necessary, round your answers to the nearest hundredth.

the vertices of $\delta abc$ are $a(2, 8)$, $b(16, 2)$, and $c(6, 2)$. the perimeter of $\delta abc$ is units, and its area is square units.

Question

type the correct answer in each box. if necessary, round your answers to the nearest hundredth.

the vertices of $\delta abc$ are $a(2, 8)$, $b(16, 2)$, and $c(6, 2)$. the perimeter of $\delta abc$ is   units, and its area is   square units.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Coordinate Geometry", "Distance Formula" ], "new_concepts": [ "Perimeter of Triangles", "Area of Triangles" ], "current_concepts": [ "Coordinate Geometry", "Distance Formula", "Perimeter of Triangles", "Area of Triangles" ] } ## Calculate side lengths of the triangle $$ \begin{aligned} AB &= \sqrt{(16 - 2)^2 + (2 - 8)^2} = \sqrt{14^2 + (-6)^2} = \sqrt{196 + 36} = \sqrt{232} \approx 15.2315\ BC &= \sqrt{(6 - 16)^2 + (2 - 2)^2} = \sqrt{(-10)^2 + 0^2} = 10\ CA &= \sqrt{(2 - 6)^2 + (8 - 2)^2} = \sqrt{(-4)^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} \approx 7.2111 \end{aligned} $$ ## Calculate perimeter of the triangle $$ \text{Perimeter} = AB + BC + CA \approx 15.2315 + 10 + 7.2111 = 32.4426 \approx 32.44 $$ ## Calculate area of the triangle $$ \begin{aligned} \text{Base } (BC) &= 10\ \text{Height } (h) &= y_A - y_B = 8 - 2 = 6\ \text{Area} &= \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 10 \times 6 = 30 \end{aligned} $$ The vertices of $\Delta ABC$ are $A(2, 8)$, $B(16, 2)$, and $C(6, 2)$. The perimeter of $\Delta ABC$ is 32.44 units, and its area is 30 square units. { "elements": [ { "type": "point", "params": [[2, 8]], "properties": { "name": "A(2,8)", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[16, 2]], "properties": { "name": "B(16,2)", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[6, 2]], "properties": { "name": "C(6,2)", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "polygon", "params": [[2, 8], [16, 2], [6, 2]], "properties": { "fillColor": "rgba(140, 85, 242, 0.2)", "strokeColor": "#8C55F2", "strokeWidth": 2 } } ] } { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Coordinate Geometry" ] }

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Explanation:

Calculate side lengths of the triangle

Calculate perimeter of the triangle

Calculate area of the triangle

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