Question

description of cross - section cross - sectional shape formed a vertical plane that cuts through the top vertex, perpendicular to the base. a horizontal plane that cuts through the pyramid, parallel to the base. a vertical plane that cuts through the base and two opposite lateral faces.

Explanation:

Step1: Analyze first description

A vertical plane through vertex, perpendicular to base: For a square pyramid (assuming base is square), a vertical plane through vertex and perpendicular to base (say, along a median of the base) will form a triangle. The right - angled triangle (if base is square and pyramid is right - square pyramid) or an isosceles triangle. Looking at the shapes, the right - angled triangle (second shape) or the equilateral - like triangle? Wait, no. If the base is a square and the pyramid is a right - square pyramid, a vertical plane through the vertex and perpendicular to the base (along the line of symmetry) will create a triangle with the same shape as the face. Wait, the first description: "A vertical plane that cuts through the top vertex, perpendicular to the base." So the cross - section should be a triangle. The fifth shape (the triangle) or the second (right - angled)? Wait, no. Let's think again. If the base is a square and the pyramid is a right square pyramid, a vertical plane through the vertex and perpendicular to the base (so it goes through the center of the base's side? No, perpendicular to the base (which is a square, so the base is in horizontal plane, perpendicular would be vertical). So the plane is vertical, through vertex, and perpendicular to base (so the intersection with the base is a line perpendicular to the base's sides). So the cross - section is a triangle. The second shape is a right - angled triangle, the fifth is a triangle. Wait, maybe the base is a square, so the vertical plane through vertex and perpendicular to base (say, along the diagonal of the base? No, perpendicular to base. Wait, the base is a square, so its sides are horizontal/vertical. A vertical plane perpendicular to the base (so the plane is vertical and its normal is horizontal, parallel to base's side). So it cuts through two opposite edges of the base? No, through vertex and perpendicular to base. So the cross - section is a triangle. Let's check the shapes: the second shape is a right - angled triangle, the fifth is a triangle. Wait, maybe the first cross - section (description 1) corresponds to the right - angled triangle (second shape) if the base is a square and the pyramid is a right - square pyramid (so the base has right angles, and the plane is perpendicular to the base, so the cross - section has a right angle).

Step2: Analyze second description

"A horizontal plane that cuts through the pyramid, parallel to the base." A horizontal plane parallel to the base of a square pyramid will create a square (or a rectangle similar to the base). The base is a square, so the cross - section should be a square - like shape. The third shape (the square with equal sides marked) or the first shape (rectangle with right angles). Wait, a horizontal plane parallel to the base of a square pyramid will produce a square (similar to the base, scaled down). So the third shape (the square with the tick marks on the sides, indicating equal sides) or the first shape (rectangle with right angles). Wait, if the base is a square, the cross - section parallel to the base is a square. So the third shape (the square with four equal sides, marked with ticks) or the first (rectangle). No, the first shape is a rectangle with right angles, the third is a square. So the second description's cross - section is a square (third shape) or a similar square - like figure.

Step3: Analyze third description

"A vertical plane that cuts through the base and two opposite lateral faces." So this plane is vertical, cuts through the base (a square) and two opposite lateral faces. So the cross - section will be a trapezoid? Wait, no. If the base is a square, and the plane cuts through two opposite lateral faces (which are triangles) and the base, the cross - section will be a trapezoid. The sixth shape (the trapezoid - like shape) or let's see. The third description: "A vertical plane that cuts through the base and two opposite lateral faces." So the cross - section has two parallel sides (from the base and the cut on the lateral faces) and two non - parallel sides (from the lateral faces). So the sixth shape (the trapezoid) or the first? No, the first is a rectangle. Wait, maybe the base is a square, and the plane is vertical, cutting through two opposite lateral faces (which are triangles) and the base. So the cross - section is a trapezoid. The sixth shape is a trapezoid.

Now let's match:

1. "A vertical plane that cuts through the top vertex, perpendicular to the base." → The cross - sectional shape is the right - angled triangle (second shape) or the triangle (fifth). Wait, maybe the fifth shape (the triangle) is the correct one for the first description. Wait, no, if the base is a square, and the plane is vertical through vertex and perpendicular to base, the cross - section is a triangle. The fifth shape is a triangle, the second is a right - angled triangle. Maybe the base is a square, so the plane perpendicular to the base (which has right angles) will create a right - angled triangle (second shape).

2. "A horizontal plane that cuts through the pyramid, parallel to the base." → The cross - sectional shape is the square (third shape) (since it's parallel to the square base, so similar square).

3. "A vertical plane that cuts through the base and two opposite lateral faces." → The cross - sectional shape is the trapezoid (sixth shape).

Wait, maybe the pyramid is a square pyramid with square base. Let's re - list:

1. Description 1: Vertical plane through vertex, perpendicular to base → Cross - section: Triangle (the fifth shape? No, the second is right - angled, fifth is a triangle. Wait, the second shape is a right - angled triangle, which makes sense if the base is a square (right angles) and the plane is perpendicular to the base (so one angle is right angle from the base's side).

2. Description 2: Horizontal plane parallel to base → Cross - section: Square (third shape, with equal sides, marked).

3. Description 3: Vertical plane through base and two opposite lateral faces → Cross - section: Trapezoid (sixth shape).

Answer:

1. For "A vertical plane that cuts through the top vertex, perpendicular to the base" → The right - angled triangle (second shape).
2. For "A horizontal plane that cuts through the pyramid, parallel to the base" → The square (third shape).
3. For "A vertical plane that cuts through the base and two opposite lateral faces" → The trapezoid (sixth shape).