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. figure (a) (b) (c) cross section various shapes with radio buttons

Explanation:

Step1: Analyze Figure (a)

Figure (a) is a sphere with a plane cutting through it. The cross - section of a sphere cut by a plane is a circle. So we look for the circle option in the cross - section row for (a).

Step2: Analyze Figure (b)

Figure (b) is a cylinder with a plane cutting it parallel to its bases? No, the plane is cutting it such that the cross - section should be a rectangle? Wait, no, when a plane cuts a cylinder parallel to its axis, the cross - section is a rectangle, but in this case, the plane is cutting the cylinder horizontally? Wait, no, the red figure is a cylinder, and the blue plane is cutting it. Wait, actually, when a plane cuts a cylinder perpendicular to its axis (i.e., parallel to the bases), the cross - section is a circle. Wait, no, the cylinder's bases are circles. If the plane is parallel to the bases, cross - section is circle; if parallel to the axis, it's a rectangle. Looking at the figure, the plane seems to be cutting the cylinder parallel to the axis? Wait, no, the red cylinder has the blue plane cutting through it. Wait, maybe I made a mistake. Wait, the cross - section options: for (b), the correct cross - section. Wait, the cylinder, when cut by a plane parallel to its axis (the line through the centers of the two circular bases), the cross - section is a rectangle. But looking at the options, there is a rectangle and a circle. Wait, no, maybe the plane is cutting the cylinder perpendicular to the axis, so cross - section is a circle? Wait, no, the cylinder's lateral surface: if you cut a cylinder with a plane perpendicular to the axis, you get a circle (the base). If you cut it with a plane parallel to the axis, you get a rectangle. Let's re - examine the figure. The red cylinder has the blue plane cutting through it, and the dashed lines suggest that the plane is parallel to the axis, so cross - section should be a rectangle? Wait, no, the options for (b) have a rectangle and a circle. Wait, maybe the figure (b) is a cylinder, and the plane is cutting it such that the cross - section is a rectangle. Wait, no, the first cross - section option for (b) is a rectangle, the second is a circle. Wait, maybe I messed up. Let's start over.

Wait, the problem is to choose the cross - section for each figure. Let's take figure (a): sphere, cross - section is circle. So in the cross - section row for (a), we select the circle.

Figure (b): cylinder. If the plane is cutting the cylinder parallel to its axis (the vertical axis, assuming the cylinder is standing on its base), then the cross - section is a rectangle. Wait, no, the cylinder's axis is the line connecting the centers of the two circular bases. If the plane is parallel to the axis, the cross - section is a rectangle. If the plane is perpendicular to the axis, it's a circle. Looking at the figure, the blue plane is cutting the cylinder in a way that the cross - section should be a rectangle? Wait, no, the red cylinder, the blue plane: maybe the cross - section is a rectangle. Wait, the first cross - section option for (b) is a rectangle, so we select that.

Figure (c): The red figure is a triangular prism? Wait, no, it's a pentagonal prism? No, it's a shape with a triangular face? Wait, the red figure has a triangular end? Wait, no, the red figure looks like a prism with a triangular base? Wait, the cross - section: when a plane cuts a triangular prism parallel to its bases (the triangular bases), the cross - section is a triangle. Wait, the last option for (c) is a triangle. So for (c), we select the triangle.

Wait, maybe I should go step by step for each figure:

Figure (a)

- The solid is a sphere.
- A plane intersecting a sphere will always form a circular cross - section (or a point, but here it's a proper intersection). So the cross - section is a circle. So in the cross - section column for (a), we choose the circle (the second option in the cross - section row for (a)).

Figure (b)

- The solid is a cylinder.
- The plane is intersecting the cylinder parallel to its axis (the line through the centers of the two circular bases). When a plane intersects a cylinder parallel to its axis, the cross - section is a rectangle. So in the cross - section column for (b), we choose the rectangle (the first option in the cross - section row for (b)).

Figure (c)

- The solid appears to be a triangular prism (with a triangular base).
- The plane is intersecting the prism parallel to its triangular bases. When a plane intersects a triangular prism parallel to its triangular bases, the cross - section is a triangle. So in the cross - section column for (c), we choose the triangle (the last option in the cross - section row for (c)).

Answer:

• For Figure (a): The circle option in its cross - section row.
• For Figure (b): The rectangle option in its cross - section row.
• For Figure (c): The triangle option in its cross - section row.

(If we need to mark the radio buttons, for (a) mark the circle, for (b) mark the rectangle, for (c) mark the triangle)