To determine the three-dimensional object formed when triangle \( \triangle ABC \) is rotated about side \( \overline{CB} \), we first need to identify the dimensions of the triangle and the axis of rotation.
Given the points:
- \( A(0,0) \)
- \( B(3,5) \)
- \( C(0,5) \)
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Identify the position of the vertices:
- Point \( C \) is located at the top of the triangle along the vertical axis (y-axis) at \( (0,5) \).
- Point \( B \) is located at \( (3,5) \), 3 units away from \( C \) horizontally.
- Point \( A \) is located at \( (0,0) \) at the base of the triangle.
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Identify the length of the segments:
- The length of \( \overline{CB} \) (the side along which the triangle is rotated) is \( 3 \) units (the distance from \( C \) to \( B \)).
- The height of the triangle from \( A \) to line \( CB \) is \( 5 \) units.
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Determine the solid of revolution:
- When the triangle is rotated around the vertical side \( \overline{CB} \), it creates a cone shape.
- The height of the cone is the distance from point \( A \) to line \( CB \), which is \( 3 \) units horizontally (since \( C \) to \( B \) is horizontal and point \( A \) is directly below \( C \)) as it extends perpendicularly, creating a cone with its vertex at \( A \) and the base along the line determined by the rotation around \( \overline{CB} \).
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Determine the base diameter:
- The diameter of the cone's base is equal to the distance from point \( B(3,5) \) to point \( C(0,5) \), which is \( 3 \) units on either side since point \( B \) is \( 3 \) units away from the y-axis.
- Thus, the base radius of the cone is \( 3 \) units, giving a diameter of \( 6 \) units (as it extends \( 3 \) units to either side of the y-axis).
Therefore, the cone formed when triangle \( \triangle ABC \) is rotated about side \( \overline{CB} \) has:
- Height: \( 3 \) units
- Diameter: \( 6 \) units
This corresponds to the first response from the options given:
A cone with a diameter of 6 units and a height of 5 units.