alex wants to fence in an area for a dog park. he has plotted three sides of the fenced area at the points e (3, 5), f (6, 5), and g (9, 1). he has 22 units of fencing. where could alex place point h so that he does not have to buy more fencing?

(0, 0)
(-1, 0)
(0, -3)
(0, 3)

Question

alex wants to fence in an area for a dog park. he has plotted three sides of the fenced area at the points e (3, 5), f (6, 5), and g (9, 1). he has 22 units of fencing. where could alex place point h so that he does not have to buy more fencing?

(0, 0)
(-1, 0)
(0, -3)
(0, 3)

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": [ "Coordinate Distance", "Coordinate Geometry Perimeter", "Rectangle Vertices" ], "new_concepts": [ "Parallelogram Vertices", "Coordinate Geometry Optimization" ], "current_concepts": [ "Coordinate Distance", "Coordinate Geometry Perimeter", "Rectangle Vertices", "Parallelogram Vertices", "Coordinate Geometry Optimization" ] } ## Calculate the lengths of the plotted sides Using the **Coordinate Distance** knowledge point $$ \begin{aligned} &E = (3, 5),\ F = (6, 5),\ G = (9, 1)\ &EF = \sqrt{(6-3)^2 + (5-5)^2} = \sqrt{3^2 + 0^2} = 3\ &FG = \sqrt{(9-6)^2 + (1-5)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = 5 \end{aligned} $$ ## Determine the required length of the remaining sides Using the **Coordinate Geometry Perimeter** knowledge point $$ \begin{aligned} &\text{Total fencing available} = 22\ &\text{Plotted sides length} = EF + FG = 3 + 5 = 8\ &\text{Remaining fencing for sides } GH \text{ and } HE = 22 - 8 = 14\ &GH + HE \le 14 \end{aligned} $$ ## Test the given coordinate options for point H Using the **Coordinate Distance** knowledge point $$ \begin{aligned} &\text{Option A: } H(0, 0)\ &GH = \sqrt{(0-9)^2 + (0-1)^2} = \sqrt{81 + 1} = \sqrt{82} \approx 9.06\ &HE = \sqrt{(3-0)^2 + (5-0)^2} = \sqrt{9 + 25} = \sqrt{34} \approx 5.83\ &GH + HE \approx 9.06 + 5.83 = 14.89 > 14 \quad (\text{Invalid})\\ &\text{Option B: } H(-1, 0)\ &GH = \sqrt{(-1-9)^2 + (0-1)^2} = \sqrt{100 + 1} = \sqrt{101} \approx 10.05\ &HE = \sqrt{(3-(-1))^2 + (5-0)^2} = \sqrt{16 + 25} = \sqrt{41} \approx 6.40\ &GH + HE \approx 10.05 + 6.40 = 16.45 > 14 \quad (\text{Invalid})\\ &\text{Option C: } H(0, -3)\ &GH = \sqrt{(0-9)^2 + (-3-1)^2} = \sqrt{81 + 16} = \sqrt{97} \approx 9.85\ &HE = \sqrt{(3-0)^2 + (5-(-3))^2} = \sqrt{9 + 64} = \sqrt{73} \approx 8.54\ &GH + HE \approx 9.85 + 8.54 = 18.39 > 14 \quad (\text{Invalid})\\ &\text{Option D: } H(0, 3)\ &GH = \sqrt{(0-9)^2 + (3-1)^2} = \sqrt{81 + 4} = \sqrt{85} \approx 9.22\ &HE = \sqrt{(3-0)^2 + (5-3)^2} = \sqrt{9 + 4} = \sqrt{13} \approx 3.61\ &GH + HE \approx 9.22 + 3.61 = 12.83 \le 14 \quad (\text{Valid}) \end{aligned} $$ ## Verify the valid location Using the **Rectangle Vertices** knowledge point $$ \begin{aligned} &\text{For } H(0, 3):\ &GH + HE \approx 12.83 \le 14\ &\text{Total perimeter } EF + FG + GH + HE \approx 3 + 5 + 9.22 + 3.61 = 20.83 \le 22 \end{aligned} $$

(A) (0, 0)

(B) (-1, 0)

(C) (0, -3)

(D) (0, 3)

{ "elements": [ { "type": "point", "params": [[3, 5]], "properties": { "name": "E (3, 5)", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[6, 5]], "properties": { "name": "F (6, 5)", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[9, 1]], "properties": { "name": "G (9, 1)", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[0, 3]], "properties": { "name": "H (0, 3)", "size": 4, "color": "#F2557F", "withLabel": true } }, { "type": "segment", "params": [[3, 5], [6, 5]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 3 } }, { "type": "segment", "params": [[6, 5], [9, 1]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 3 } }, { "type": "segment", "params": [[9, 1], [0, 3]], "properties": { "strokeColor": "#F28A55", "strokeWidth": 2, "dash": 2 } }, { "type": "segment", "params": [[0, 3], [3, 5]], "properties": { "strokeColor": "#F28A55", "strokeWidth": 2, "dash": 2 } } ] } { "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Coordinate Geometry Perimeter" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate the lengths of the plotted sides

Determine the required length of the remaining sides

Test the given coordinate options for point H

Verify the valid location

Answer: