Question

the diagram shows a cube of side length 3 inches. the points p, q, and r have the relative locations shown. p is one of the vertices on the bottom of the cube, q is located on an edge and is 1 inch from the top of the cube, and r is located on an edge and is 1 inch from the bottom of the cube.

if a plane including the points p, q, and r cuts through the cube, what is the shape of the resulting cross section?

Explanation:

Step1: Assign coordinates to cube vertices

Let the cube have side length 3. Place P at (3,3,0) (bottom vertex). Q is 1 in from top on edge: (3,2,3) (since top z=3, so z=3-1=2? Correction: Wait, standard cube: let’s set origin at bottom-front-left. Let P be (3,3,0) (bottom-back-right vertex). Q is on right edge, 1 in from top: right edge has x=3,y=3,z from 0 to3, so Q=(3,3,2) (z=3-1=2). R is on front edge, 1 in from bottom: front edge x=0,y=3,z from0 to3, so R=(0,3,1) (z=1).

Step2: Analyze plane cutting cube

A plane through three non-collinear points in a cube cuts edges based on their positions. The plane through P(3,3,0), Q(3,3,2), R(0,3,1) intersects multiple edges of the cube. The intersection will form a polygon with 5 sides because the plane crosses 5 edges of the cube.

Answer:

Pentagon