Question

math 132, w01: review
directions: respond to each question in the space provided below. show all mathematical steps used to arrive at your answer, and justify all your steps by citing the appropriate theorems, definitions, or properties. you may use a calculator and protractor on this exam. no notes, computers, mobile devices, or other electronic devices may be used on this exam.

1. the following figure is a box in which the top and the bottom are rectangles with (overline{bf}) and (overline{dh}) perpendicular to the planes fgh and bcd.

a. find the intersection of (overline{bh}) (not shown in picture) and plane dcg.
b. name two pairs of perpendicular planes.
c. name the intersection of planes aeh and dcg.

Explanation:

Step1: Analyze line - plane intersection for part a

A line intersects a plane at a point. In the given box, $\overline{BH}$ intersects plane $DCG$ at point $D$. This is because $\overline{BH}$ passes through the corner - point $B$ and extends through the box, and the only common point with plane $DCG$ is the point where the line enters the plane, which is $D$.

Step2: Identify perpendicular planes for part b

By the properties of a rectangular - box, planes that form right - angles with each other are perpendicular. Planes $ABCD$ and $BCGF$ are perpendicular since the edges between them are at right - angles. Also, planes $EFGH$ and $DCGH$ are perpendicular as the edges connecting them are perpendicular.

Step3: Determine plane - plane intersection for part c

The intersection of two non - parallel planes is a line. Planes $AEH$ and $DCG$ intersect along the line $\overline{HG}$. This is because $\overline{HG}$ is the common line that lies on both planes $AEH$ (since $H$ and $G$ are points on it) and $DCG$ (as $H$ and $G$ are points on this plane too).

Answer:

a. Point $D$
b. Planes $ABCD$ and $BCGF$, Planes $EFGH$ and $DCGH$
c. Line $\overline{HG}$