Question

in each figure below, a plane passes through a geometric solid as shown. for each figure, choose the shape of the cross section formed.

Explanation:

Figure (a) (Cylinder)

Step1: Analyze the cylinder and the plane

A cylinder has two circular bases and a curved lateral surface. The plane in figure (a) is parallel to the axis of the cylinder (the line connecting the centers of the two circular bases) and is cutting through the cylinder.

Step2: Determine the cross - section shape

When a plane parallel to the axis of a cylinder cuts through it, the intersection with the cylinder will form a rectangle. The two vertical sides of the rectangle are formed by the intersection of the plane with the lateral surface of the cylinder, and the top and bottom sides are formed by the intersection of the plane with the two circular bases (or the extensions of the bases).

Figure (b) (Cone)

Step1: Analyze the cone and the plane

A cone has a circular base and a pointed apex. The plane in figure (b) is passing through the apex of the cone and is also cutting through the circular base.

Step2: Determine the cross - section shape

When a plane passes through the apex of a cone and intersects the base, the cross - section formed is a triangle. The two sides of the triangle are formed by the intersection of the plane with the lateral surface of the cone, and the base of the triangle is formed by the intersection of the plane with the circular base of the cone.

Figure (c) (A combination of a prism and a pyramid)

Step1: Analyze the solid and the plane

The solid in figure (c) has a rectangular - prism - like part and a pyramid - like part. The plane is cutting through both parts. On the prism part, the intersection with the plane will be a rectangle - like shape, and on the pyramid part, the intersection will be a triangle - like shape. Combining these, we get a trapezoid.

Step2: Determine the cross - section shape

A trapezoid has one pair of parallel sides. The side from the prism part and the side from the pyramid part (after the plane cuts through) are parallel, and the other two sides are non - parallel, which matches the shape of a trapezoid.

Answer:

The cross - section shape is a rectangle (the last option in the cross - section row for figure (a)).