question 10 of 10

is the following shape a square? how do you know?

a. no, the sides are not congruent.
b. no, opposite sides are not parallel.
c. no, adjacent sides are not perpendicular.
d. yes, the opposite sides are parallel, and all sides are the same length.

Question

question 10 of 10

is the following shape a square? how do you know?

a. no, the sides are not congruent.
b. no, opposite sides are not parallel.
c. no, adjacent sides are not perpendicular.
d. yes, the opposite sides are parallel, and all sides are the same length.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Distance Formula", "Perpendicular Slopes", "Coordinate Geometry" ], "new_concepts": [ "Properties of a Square" ], "current_concepts": [ "Distance Formula", "Perpendicular Slopes", "Coordinate Geometry", "Properties of a Square" ] } ## Identify coordinates of the vertices $$ A = (-1, 2), \quad B = (-3, 0), \quad C = (-1, -2), \quad D = (1, 0) $$ ## Calculate side lengths using the distance formula $$ \begin{aligned} AB &= \sqrt{(-1 - (-3))^2 + (2 - 0)^2} = \sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2} \ BC &= \sqrt{(-3 - (-1))^2 + (0 - (-2))^2} = \sqrt{(-2)^2 + 2^2} = \sqrt{8} = 2\sqrt{2} \ CD &= \sqrt{(-1 - 1)^2 + (-2 - 0)^2} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{8} = 2\sqrt{2} \ DA &= \sqrt{(1 - (-1))^2 + (0 - 2)^2} = \sqrt{2^2 + (-2)^2} = \sqrt{8} = 2\sqrt{2} \end{aligned} $$ ## Calculate slopes of adjacent sides to check perpendicularity $$ \begin{aligned} m_{AB} &= \frac{2 - 0}{-1 - (-3)} = \frac{2}{2} = 1 \ m_{BC} &= \frac{0 - (-2)}{-3 - (-1)} = \frac{2}{-2} = -1 \ m_{AB} \cdot m_{BC} &= 1 \cdot (-1) = -1 \end{aligned} $$

A. No, the sides are not congruent.

B. No, opposite sides are not parallel.

C. No, adjacent sides are not perpendicular.

D. Yes, the opposite sides are parallel, and all sides are the same length.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Properties of a Square" ] }

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QUESTION IMAGE

Question

Explanation:

Identify coordinates of the vertices

Calculate side lengths using the distance formula

Calculate slopes of adjacent sides to check perpendicularity

Answer: