Question

a2 name:
lesson 2 practice problems

1. select all figures for which there exists a direction such that all cross sections taken at that direction are congruent.

a. triangular pyramid
b. square pyramid
c. rectangular prism
d. cube
e. cone
f. cylinder
g. sphere

Explanation:

To determine which figures have all cross - sections (in a certain direction) congruent, we analyze each figure:

• Triangular Pyramid (A): Cross - sections parallel to the base are triangles, but cross - sections in other directions (e.g., parallel to a face) will have different shapes and sizes. So, not all cross - sections are congruent.
• Square Pyramid (B): Similar to the triangular pyramid, cross - sections parallel to the base are squares, but cross - sections in other directions will vary. Not all cross - sections are congruent.
• Rectangular Prism (C): If we take cross - sections parallel to the faces (in the direction perpendicular to the length, width, or height), the cross - sections will be rectangles (or squares) that are congruent. For example, if we take cross - sections parallel to the front - back face, they will all have the same length and height.
• Cube (D): A cube is a special case of a rectangular prism. All cross - sections taken parallel to any of its faces (in the direction perpendicular to the face) will be squares of the same size. So, all cross - sections in those directions are congruent.
• Cone (E): Cross - sections parallel to the base are circles, but cross - sections in the direction from the vertex to the base (non - parallel to the base) are triangles, and their sizes change as we move from the vertex to the base. Not all cross - sections are congruent.
• Cylinder (F): If we take cross - sections parallel to the circular bases (in the direction perpendicular to the height of the cylinder), the cross - sections will be circles of the same radius. So, all cross - sections in that direction are congruent.
• Sphere (G): Any cross - section of a sphere (a plane cutting through the sphere) is a circle, and all these circles have the same radius (since the sphere is symmetric in all directions). So, all cross - sections of a sphere are congruent.

Answer:

C. rectangular prism, D. cube, F. cylinder, G. sphere