۲۴ در شکل مقابل، فاصله نقطه a از مبدأ مختصات کدام است؟
۱) (2sqrt{5})
۲) (3sqrt{6})
۳) (4sqrt{3})
۴) (5sqrt{2})

Question

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [], "new_concepts": [ "Coordinate Geometry", "Line Equations", "Distance Formula" ], "current_concepts": [ "Coordinate Geometry", "Line Equations", "Distance Formula" ] } ## Find the equation of the line $$ \begin{aligned} &\theta = 180^\circ - 135^\circ = 45^\circ \implies m = \tan(180^\circ - 45^\circ) = -1\ &y\text{-intercept} = 2 \implies y = -x + 2 \implies x + y - 2 = 0 \end{aligned} $$ ## Determine the coordinates of point A $$ \begin{aligned} &x\text{-intercept: } y = 0 \implies x = 2 \implies P(2, 0)\ &\text{Since } P \text{ is the midpoint between } (0, 2) \text{ and } A(x_A, y_A):\ &2 = \frac{0 + x_A}{2} \implies x_A = 4\ &0 = \frac{2 + y_A}{2} \implies y_A = -2 \implies A(4, -2) \end{aligned} $$ ## Calculate the distance from the origin to A $$ d = \sqrt{4^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} $$

(1) $2\sqrt{5}$

(2) $3\sqrt{6}$

(3) $4\sqrt{3}$

(4) $5\sqrt{2}$

{ "elements": [ { "type": "line", "params": [[0, 2], [4, -2]], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2 } }, { "type": "point", "params": [[0, 2]], "properties": { "name": "(0,2)", "size": 4, "color": "#5583F2", "withLabel": true } }, { "type": "point", "params": [[2, 0]], "properties": { "name": "(2,0)", "size": 4, "color": "#5583F2", "withLabel": true } }, { "type": "point", "params": [[4, -2]], "properties": { "name": "A(4,-2)", "size": 4, "color": "#F2557F", "withLabel": true } }, { "type": "segment", "params": [[0, 0], [4, -2]], "properties": { "strokeColor": "#F28A55", "strokeWidth": 1.5, "dash": 2 } } ] } { "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Coordinate Geometry" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Find the equation of the line

Determine the coordinates of point A

Calculate the distance from the origin to A

Answer: