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:

Step1: Analyze Figure (a)

The solid in (a) is a sphere, and the plane is cutting through it. A plane cutting a sphere will form a circle or a semicircle? Wait, no—wait, the cross - section of a sphere cut by a plane is always a circle. Wait, but looking at the options, the last row has circles. Wait, no, the first cross - section row for (a): the plane is cutting the sphere, so the cross - section should be a circle. Wait, but let's check again. Wait, the figure (a) shows a sphere with a plane cutting through it, so the cross - section is a circle. But in the options for (a), the last row (the sixth row) has circles. Wait, maybe I misread. Wait, no, let's take each figure one by one.

Step2: Analyze Figure (b)

The solid in (b) is a cone, and the plane is cutting through it parallel to the base? No, the plane is cutting through the cone. Wait, if the plane cuts the cone such that it's parallel to the base, the cross - section is a circle, but if it's at an angle, it's an ellipse or a triangle. Wait, no, the figure (b) shows a cone with a plane cutting through it. Wait, the cross - section options: the first row for (b) has a triangle, but a plane cutting a cone through the vertex and the base would form a triangle. Wait, no, the plane in (b) is cutting the cone, and the cross - section should be a triangle? Wait, no, maybe I'm wrong. Wait, let's look at figure (c): it's a cylinder, and the plane is cutting through it parallel to the bases? No, the plane is cutting through the cylinder, so the cross - section should be a rectangle (if the plane is parallel to the axis) or a circle (if parallel to the base). Wait, the cylinder in (c) has a plane cutting through it, and the cross - section options for (c) have a rectangle in the fourth row.

Wait, maybe the correct cross - sections are:

For (a): The sphere is cut by a plane, so the cross - section is a circle (last row, middle or right? Wait, the figure (a) has a plane cutting the sphere, so the cross - section is a circle. So for (a), the correct cross - section is the circle in the last row (the sixth row) of (a)'s column.

For (b): The cone is cut by a plane, and if the plane cuts through the vertex and the base, the cross - section is a triangle. Wait, the first row of (b)'s column has a triangle. So for (b), the correct cross - section is the triangle in the first row.

For (c): The cylinder is cut by a plane, and if the plane is parallel to the axis (the vertical axis), the cross - section is a rectangle. The fourth row of (c)'s column has a rectangle.

But the question is to choose the shape of the cross - section for each figure. Let's re - examine:

Figure (a): Sphere, plane cutting through it. The cross - section is a circle. So in the (a) column, the last row (the one with circles) is the correct cross - section.

Figure (b): Cone, plane cutting through it. If the plane passes through the vertex and the base, the cross - section is a triangle. So in the (b) column, the first row (triangle) is correct? Wait, no, a plane cutting a cone through the vertex and the base (i.e., containing the axis) will form a triangle. So for (b), the cross - section is a triangle (first row, middle column? Wait, the (b) column's first row has a triangle.

Figure (c): Cylinder, plane cutting through it. If the plane is parallel to the axis (the line connecting the centers of the two circular bases), the cross - section is a rectangle. So in the (c) column, the fourth row (rectangle) is correct.

But maybe the intended answers are:

(a): The cross - section is a circle (last row, (a) column)

(b): The cross - section is a triangle (first row, (b) column)

(c): The cross - section is a rectangle (fourth row, (c) column)

But since the problem is about cross - sections of 3D solids, which is a Geometry (subfield of Mathematics) problem.

Answer:

For (a): The circle in the last row (6th row) of the (a) column.

For (b): The triangle in the first row of the (b) column.

For (c): The rectangle in the fourth row of the (c) column.

(If we have to choose one figure, let's assume we are to answer for all three. But the problem says "for each figure", so we provide the cross - section for each:

(a): The circle (last row, (a) column)

(b): The triangle (first row, (b) column)

(c): The rectangle (fourth row, (c) column))