Question

1. the figure below is a rectangular shipping box. name two different planes that contain \overleftrightarrow{ae}.

a. plane abf, plane dgh
b. plane abf, plane adh
c. plane efg, plane adh
d. plane dgh, plane efg

Explanation:

Step1: Analyze the line \(\overleftrightarrow{AE}\)

In the rectangular box, \(\overleftrightarrow{AE}\) is an edge. We need to find planes that contain this line. A plane containing \(\overleftrightarrow{AE}\) must have two points from \(\overleftrightarrow{AE}\) (or the line itself) and other points forming the plane.

Step2: Check each option

• Option a: Plane \(ABF\) contains \(A\) and \(E\)? No, \(E\) is not in plane \(ABF\) (plane \(ABF\) has points \(A, B, F\)). Plane \(DGH\) does not contain \(A\) or \(E\). Eliminate a.
• Option b: Plane \(ABF\) contains \(A\) and \(F\) (and \(B\)), and \(\overleftrightarrow{AE}\) is connected to \(A\) and \(E\), and \(F\) is related to the vertical edge. Wait, actually, plane \(ABF\) has \(A\), and plane \(ADH\): Wait, no, let's re - check. Wait, plane \(ABF\): points \(A, B, F\). \(\overleftrightarrow{AE}\) has \(A\), and \(E\) is below \(F\)? Wait, maybe I made a mistake. Wait, the rectangular box: \(A\) is connected to \(D\), \(B\), and \(F\) (via vertical edge). \(E\) is connected to \(F\), \(H\), \(G\). So \(\overleftrightarrow{AE}\) is a vertical edge? Wait, no, \(A\) to \(E\): maybe front - left vertical edge. Plane \(ABF\): contains \(A\) (from \(\overleftrightarrow{AE}\)) and \(F\) (connected to \(E\)). Plane \(ADH\): Wait, no, plane \(ABF\) and plane \(AEF\)? Wait, maybe the correct way is: Plane \(ABF\) contains \(A\) and the vertical edge related to \(AE\), and plane \(ADH\) – no, wait, let's check the other options. Wait, maybe the correct option is b? Wait, no, let's re - examine. Wait, the line \(\overleftrightarrow{AE}\): points \(A\) and \(E\). Plane \(ABF\) has \(A\) and \(F\) (and \(B\)), and \(F\) is adjacent to \(E\) (since \(EF\) is an edge). Plane \(ADH\): No, \(E\) is not in plane \(ADH\). Wait, maybe I messed up the labels. Wait, the box: let's assume the bottom face is \(EFGH\) (with \(E, F, G, H\)) and top face \(ABCD\) (with \(A, B, C, D\)), and vertical edges \(AE, BF, CG, DH\). So \(\overleftrightarrow{AE}\) is the edge from \(A\) (top front - left) to \(E\) (bottom front - left). Plane \(ABF\): contains \(A\) (top front - left), \(B\) (top front - right), \(F\) (bottom front - left). So \(\overleftrightarrow{AE}\) is in plane \(ABF\) (since \(A\) and \(F\) are in the plane, and \(E\) is below \(F\), so the line \(AE\) is in plane \(ABF\)). Plane \(ADH\): No, \(H\) is bottom back - right. Wait, maybe the other plane is plane \(AEH\)? No, the option b is plane \(ABF\) and plane \(ADH\) – no, maybe a typo? Wait, no, let's check option b again. Wait, maybe the correct answer is b. Wait, no, let's check the other options. Option c: Plane \(EFG\) is the bottom face? No, \(A\) is not in plane \(EFG\). Option d: Plane \(DGH\) and plane \(EFG\) do not contain \(A\). So the only option where one of the planes contains \(A\) (from \(\overleftrightarrow{AE}\)) is option b (plane \(ABF\) contains \(A\)). Wait, maybe the second plane is plane \(AEH\) or something, but among the options, option b is the only one where one plane contains \(A\) (from \(\overleftrightarrow{AE}\)). So the correct option is b.

Answer:

b. plane \(ABF\), plane \(ADH\)