نقاط $a(0, 1)$ و $b(4, -2)$ دو رأس مجاور مربع $abcd$ هستند. طول مختصات نقطه $d$ در ربع سوم، کدام است؟
-1 (1
-2 (2
-3 (3
-4 (4

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [], "new_concepts": [ "Square Coordinates", "Vector Rotation" ], "current_concepts": [ "Square Coordinates", "Vector Rotation", "Coordinate Geometry" ] } ## Find the vector AB and its perpendicular vectors $$ \begin{aligned} &\vec{AB} = B - A = (4 - 0, -2 - 1) = (4, -3)\ &\vec{u}_1 = (3, 4)\ &\vec{u}_2 = (-3, -4) \end{aligned} $$ ## Calculate the possible coordinates for D $$ \begin{aligned} &D_1 = A + \vec{u}_1 = (0 + 3, 1 + 4) = (3, 5)\ &D_2 = A + \vec{u}_2 = (0 - 3, 1 - 4) = (-3, -3) \end{aligned} $$ ## Identify the point in the third quadrant and its x-coordinate $$ \begin{aligned} &D = (-3, -3) \in \text{Quadrant III}\ &x_D = -3 \end{aligned} $$

(1) $-1$

(2) $-2$

(3) $-3$

(4) $-4$

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Explanation:

Find the vector AB and its perpendicular vectors

Calculate the possible coordinates for D

Identify the point in the third quadrant and its x-coordinate

Answer: