A triangular pyramid is cut by a plane. In which case is the cross section a quadrilateral?
When the plane is perpendicular to the base and passes through the apex
When the plane is parallel to the base of the pyramid
When the plane intersects the base and two lateral faces of the pyramid
When the plane intersects the base and all three lateral faces of the pyramid

Solution:

The cross section will be a quadrilateral when the intersecting plane forms a closed four sided polygon along with the pyramid.

A triangular pyramid is simply defined as a tetrahedron or regular tetrahedron with a triangular base.

In the tetrahedron all edges are equal. From the above figure, all edges AB = BC = AC = AS = BS = CS.

In a regular tetrahedron, all faces and bases are equilateral triangles.

A triangular pyramid has four faces. To form a quadrilateral, the intersecting plane must intersect all four faces.

From options given in question,

1. when the plane is perpendicular to the base and passes through the apex, the plane will intersect only one side of the pyramid. So, a four sided quadrilateral can’t be formed.
2. When the plane is parallel to the base of the pyramid, the plane will intersect two sides of the pyramid. Hence the cross section will not be a quadrilateral.
3. when the plane intersects the base and two lateral faces of the pyramid, the plane will intersect three sides of the pyramid. So the cross section will be a triangle.
4. When the plane intersects the base and all three lateral faces of the pyramid, the plane will intersect all the four faces of the pyramid thereby forming a closed polygon of four straight lines. So the cross section will be a quadrilateral.

Therefore, when the plane intersects the base and all three lateral faces of the pyramid, the cross section formed will be a quadrilateral.

A triangular pyramid is cut by a plane. In which case is the cross section a quadrilateral?

Summary:

A triangular pyramid cut by a plane that intersects the base and all three lateral faces, the cross section so formed will be a quadrilateral.