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A figure is located at (2,0), (2,-2), and (6,0) on a coordinate plane. What kind of 3-D shape would be created if the figure was rotated around the x-axis? Provide an explanation and proof of your answer to receive full credit. Include the dimensions of the 3-D shape in your explanation.

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Final answer:

A figure with vertices at (2,0), (2,-2), and (6,0), when rotated around the x-axis, results in a cone with a base radius of 2 units and a height of 2 units. This process of forming a 3-D shape through rotation is known as revolution in the field of geometry.

Step-by-step explanation:

When a figure with vertices at (2,0), (2,-2), and (6,0) on a two-dimensional coordinate plane is rotated around the x-axis, a three-dimensional shape known as a cone is formed. Imagine dragging the triangle through space along the x-axis to visualize the resulting solid. This method of creating a shape is known as revolution, which is a type of transformation in geometry.

The dimensions of the cone can be determined by examining the triangle. The base of the cone will be a circle formed by the segment between (2,-2) and (6,0) when it is rotated around the x-axis. The radius of this circle is 2 units because the y-coordinates of the endpoints signify the distance from the x-axis. The height of the cone can be determined by the difference in x-coordinates of the line segment that represents the height in the triangle, which is from (2,0) to (2,-2), resulting in a height of 2 units.

To summarize, the triangle formed by the given points rotates around the x-axis to create a cone with a base radius of 2 units and a height of 2 units. The dimensions are derived from the length of the sides of the triangle in the coordinate system, following the right-hand rule for Cartesian coordinates.

User Art Geigel
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