which of the following statements is true for the rotation of any shape?

- the number of different cross sections of the resulting solid will be equal to the number of sides of the shape that was rotated.
- the number of faces on the resulting solid will be equal to the number of sides on the shape that was rotated.
- the number of edges on the resulting solid will be equal to three times the number of sides on the shape that was rotated.
- if the shape was rotated about a line of symmetry, there will be at least one cross section of the resulting solid that produces the original shape.

Question

which of the following statements is true for the rotation of any shape?

- the number of different cross sections of the resulting solid will be equal to the number of sides of the shape that was rotated.
- the number of faces on the resulting solid will be equal to the number of sides on the shape that was rotated.
- the number of edges on the resulting solid will be equal to three times the number of sides on the shape that was rotated.
- if the shape was rotated about a line of symmetry, there will be at least one cross section of the resulting solid that produces the original shape.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "direct_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Solids of Revolution", "Cross Sections of 3D Shapes" ], "new_concepts": [], "current_concepts": [ "Solids of Revolution", "Cross Sections of 3D Shapes" ] } ## Analyze the geometric properties of solids of revolution $$ \begin{aligned} &\text{Let } S \text{ be a 2D shape rotated } 360^\circ \text{ about an axis } L \text{ to form a 3D solid } V.\ &\text{If } L \text{ is a line of symmetry of } S, \text{ then } S \text{ is symmetric with respect to } L.\ &\text{A plane containing the axis of rotation } L \text{ intersects } V \text{ to form a cross section.}\ &\text{This cross section is the union of } S \text{ and its reflection across } L.\ &\text{Since } L \text{ is a line of symmetry of } S, \text{ the union is exactly the original shape } S. \end{aligned} $$ ## Evaluate the given statements $$ \begin{aligned} &\text{Option 1: False. A rotated triangle (3 sides) yields a cone, which has infinitely many distinct cross sections.}\ &\text{Option 2: False. A rotated circle (1 boundary) yields a sphere (1 face), but a rotated rectangle (4 sides) yields a cylinder (3 faces).}\ &\text{Option 3: False. A rotated rectangle yields a cylinder, which has 2 curved edges, not } 3 \times 4 = 12 \text{ edges.}\ &\text{Option 4: True. The cross section along the plane containing the axis of symmetry yields the original shape.} \end{aligned} $$

(A) The number of different cross sections of the resulting solid will be equal to the number of sides of the shape that was rotated.

(B) The number of faces on the resulting solid will be equal to the number of sides on the shape that was rotated.

(C) The number of edges on the resulting solid will be equal to three times the number of sides on the shape that was rotated.

(D) If the shape was rotated about a line of symmetry, there will be at least one cross section of the resulting solid that produces the original shape.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Solids of Revolution" ] }

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Question

Explanation:

Analyze the geometric properties of solids of revolution

Evaluate the given statements

Answer: