Question

select the correct answer.
a prism whose base is a 9 - sided polygon is intersected by a plane. in which case could the cross section be a 10 - sided polygon?
a. when the plane is parallel to the base of the prism
b. when the plane is perpendicular to the base and contains two edges of the prism
c. when the plane is perpendicular to the base and contains one edge of the prism
d. when the plane makes a narrow angle with the base

Explanation:

Step1: Analyze Option A

If the plane is parallel to the base (a 9 - sided polygon), the cross - section will be congruent to the base, so it will also be a 9 - sided polygon. So option A is incorrect.

Step2: Analyze Option B

The base of the prism is a 9 - sided polygon. A prism has two congruent polygonal bases and rectangular (or parallelogram - shaped) lateral faces. If the plane is perpendicular to the base and contains two edges of the prism:
- The base has 9 sides. When the plane intersects the prism, it will intersect the two bases (each intersection gives a side) and the lateral faces. Since it contains two edges, the number of sides of the cross - section: the base has 9 sides, but when the plane is perpendicular and contains two edges, the cross - section will have 9 + 1=10 sides? Wait, let's think again. A prism with a 9 - sided base (non -agonal prism) has 9 lateral edges. The cross - section is formed by the intersection of the plane with the faces of the prism.
- When the plane is perpendicular to the base and contains two edges: The plane will intersect the two bases (each intersection is a line segment) and the lateral faces. The number of sides of the cross - section: the base is 9 - sided. If the plane contains two edges, the number of sides from the lateral faces and the bases: Let's consider the general formula for the number of sides of a cross - section of a prism. For a prism with an \(n\) - sided base, when a plane intersects the prism:
- If the plane is parallel to the base, the cross - section is \(n\) - sided.
- If the plane is perpendicular to the base and intersects \(k\) lateral edges, the number of sides of the cross - section is \(n + k\) (in some cases). Wait, maybe a better way: A 9 - sided prism (enneagonal prism) has 9 lateral faces (rectangles) and 2 enneagonal bases.
- When the plane is perpendicular to the base and contains two edges: The plane will intersect 9 lateral faces? No, wait. Let's take a simple case: a triangular prism (3 - sided base). If a plane is perpendicular to the base and contains two edges, the cross - section is a rectangle? No, maybe my initial approach is wrong.
- Wait, the base is 9 - sided, so the prism has 9 lateral edges (connecting the corresponding vertices of the two bases). The cross - section is a polygon whose sides are the intersections of the plane with the faces of the prism.
- For option B: The plane is perpendicular to the base (so it is parallel to the lateral edges) and contains two edges. The number of sides of the cross - section: the base has 9 sides. When the plane contains two edges, the cross - section will have 9+1 = 10 sides? Wait, no. Let's think of the prism as having 9 lateral faces (each is a rectangle) and 2 bases (9 - sided). When the plane is perpendicular to the base and contains two edges, the plane will intersect the two bases (each intersection is a line segment, so 2 sides) and 8 lateral faces? No, this is getting confusing. Let's analyze other options.

Step3: Analyze Option C

If the plane is perpendicular to the base and contains one edge: The cross - section will be a rectangle (if the base is a polygon and the plane contains one edge and is perpendicular to the base) or a polygon with 9 + 0=9 sides? Wait, no. If the plane contains one edge, the cross - section will have a side along that edge and intersect the base (9 - sided) and some lateral faces. The number of sides will be 9 (from the base) + 0 (since it contains one edge, the intersection with the lateral faces will be along that edge) = 9? So option C is incorrect.

Step4: Analyze Option D

If the plane makes a narrow angle with the base: The cross - section will be a polygon, but the number of sides: when the plane is not parallel or perpendicular, the number of sides. Wait, for a 9 - sided base, if the plane is at an angle, the number of sides of the cross - section: Let's think of a hexagonal prism (6 - sided base). If a plane is at an angle to the base, the cross - section can have more than 6 sides? No, actually, when the plane is at an angle, the number of sides of the cross - section is equal to the number of sides of the base (if it intersects all the lateral faces and the two bases). Wait, no. I think I made a mistake in option B. Let's re - evaluate:

The base is 9 - sided, so the prism has 9 lateral edges. The cross - section is formed by the intersection of the plane with the 2 bases and the 9 lateral faces.

- Option A: Plane parallel to base: intersects 2 bases (each intersection is a 9 - sided polygon) and 0 lateral faces (since it's parallel). So cross - section is 9 - sided.
- Option B: Plane perpendicular to base (so parallel to lateral edges) and contains 2 edges: The plane intersects 2 bases (each intersection is a line segment, so 2 sides) and 9 - 2=7 lateral faces? No, the lateral faces are rectangles. If the plane is perpendicular to the base and contains 2 edges, the number of sides of the cross - section: the base has 9 sides. The plane will intersect 9 lateral faces? No, the two edges are part of the lateral edges. Wait, maybe the correct way is:

A prism with an \(n\) - sided base has \(n\) lateral edges. The cross - section of a prism, when a plane intersects it, has a number of sides equal to the number of faces of the prism that the plane intersects.

A 9 - sided prism has \(9 + 2=11\) faces (9 lateral, 2 bases).

- For option A: Plane parallel to base: intersects 2 bases (so 2 faces) and 0 lateral faces. Wait, no, the cross - section is parallel to the base, so it's a 9 - sided polygon (intersecting 2 bases and 0 lateral faces? No, that can't be. I think my face - counting is wrong.

Wait, let's start over. The problem is about a prism with a 9 - sided base (enneagonal prism). We need to find when the cross - section is 10 - sided.

A cross - section of a prism is a polygon whose vertices are the intersections of the plane with the edges of the prism.

- Option A: Plane parallel to base. The plane will intersect 9 lateral edges (the vertical edges, if the base is horizontal) at 9 points, and the two bases at 9 points each? No, if it's parallel to the base, it will intersect the two bases (each is 9 - sided) and no lateral edges. So the cross - section is 9 - sided (same as base). So A is out.

- Option B: Plane perpendicular to base and contains two edges. The plane is perpendicular to the base, so it's parallel to the lateral edges. If it contains two lateral edges, then the plane will intersect the two bases (each at a line segment, so 2 points) and 9 - 2=7 lateral edges? No, the number of vertices of the cross - section: the base has 9 vertices. The plane contains two edges (lateral edges), so it passes through two vertices of the top base and two vertices of the bottom base? Wait, no. A lateral edge connects a vertex of the bottom base to a vertex of the top base.

Wait, maybe a better approach: The number of sides of the cross - section is equal to the number of faces of the prism that the plane intersects. A prism with an \(n\) - sided base has \(n + 2\) faces (\(n\) lateral faces and 2 bases).

- For option A: Plane parallel to base. It intersects 2 faces (the two bases), so the cross - section is \(n\) - sided (9 - sided).

- For option B: Plane perpendicular to base and contains two edges. The plane is perpendicular to the base, so it is parallel to the lateral edges. It will intersect \(n+2\) faces? No, if it contains two edges, it will intersect \(n\) lateral faces and 2 bases? Wait, no. Let's take \(n = 9\). If the plane is perpendicular to the base and contains two edges, the number of faces it intersects: the two bases (2 faces) and \(n - 2=7\) lateral faces? No, this is not working.

Wait, the key is that to get a 10 - sided cross - section from a 9 - sided base, we need the plane to intersect one more face than the number of sides of the base.

- Option B: If the plane is perpendicular to the base and contains two edges, the number of sides of the cross - section: the base has 9 sides. When the plane contains two edges, it will intersect 9 + 1=10 faces? No, the number of sides of the cross - section is equal to the number of edges of the prism that the plane intersects.

A 9 - sided prism has \(9\times3 = 27\) edges? No, a prism has \(3n\) edges: \(n\) edges for the top base, \(n\) edges for the bottom base, and \(n\) lateral edges. So 27 edges for \(n = 9\).

The cross - section is a polygon with \(k\) sides, where \(k\) is the number of edges of the prism intersected by the plane.

- Option A: Plane parallel to base. It intersects \(n\) lateral edges (9 edges) and 0 base edges. So \(k = 9\) (9 - sided).

- Option B: Plane perpendicular to base and contains two edges. The plane contains two lateral edges (so it intersects 2 lateral edges) and intersects the base edges. The number of base edges intersected: the base has 9 edges. The plane is perpendicular to the base, so it intersects each base (top and bottom) at a line segment, which intersects 9 base edges? No, this is too complicated.

Wait, let's look at the answer options again. The correct answer is B? Wait, no, let's think of a simpler case: a triangular prism (3 - sided base). When is the cross - section 4 - sided? When the plane is perpendicular to the base and contains two edges. The triangular prism has 3 - sided base, cross - section 4 - sided (3 + 1). So by analogy, a 9 - sided base, cross - section 10 - sided (9+1) when the plane is perpendicular to the base and contains two edges (so 9 + 1=10). So option B is correct.

Step5: Analyze Option C

Plane perpendicular to base and contains one edge. Then the cross - section would be 9+0 = 9 - sided (like the triangular prism: plane perpendicular to base and contains one edge, cross - section is 3 - sided, same as base). So C is out.

Step6: Analyze Option D

Plane makes a narrow angle with the base. If the plane is at an angle, the number of sides of the cross - section: it will intersect more lateral edges, but the number of sides would be more than 9, but how? Wait, if the plane is at an angle, it will intersect 9 lateral edges and some base edges. But the problem is that when the plane is at an angle, the cross - section is a non - regular polygon, but the number of sides: for a 9 - sided base, if the plane is at an angle, it will intersect 9 lateral edges and 2 base edges (one from top base, one from bottom base), so 11 sides? No, that doesn't make sense. So D is out.

Answer:

B. when the plane is perpendicular to the base and contains two edges of the prism