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Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698

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Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698
Textbook Problem
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A hexagonal pyramid (not regular) with base A B C D E F has plane symmetry with respect to a plane determined by apex G and vertices A and D of its base. If the volume of the pyramid with apex G and base A B C D is 19.7 i n 3 , find the volume of the given hexagonal pyramid.

To determine

To find:

The volume of the pyramid.

Explanation

Given:

A hexagonal pyramid (not regular) with base ABCDEF has plane symmetry with respect to a plane determined by apex G and vertices A and D of its base

and the volume of the pyramid with apex G and base ABCD is 19.7in3.

Properties Used:

Symmetry is when two or more parts are identical after a flip, slide or turn. A plane of symmetry is an imaginary plane along which the figure is flipped, slides or turned to get two or more identical parts.

To have this type of symmetry, a plane can be drawn for which each point of the space figure has a corresponding point on the opposite side of the plane at the same distance from the plane as shown in figure below.

The plane of symmetry divides the pyramid in two equal solids with the same volume.

Calculation:

Since, in the hexagonal pyramid (not regular) with base ABCDEF has plane symmetry with respect to a plane determined by apex G and vertices A and D of its base.

Hence, the volume of the whole pyramid is

V=VGABCD+VGDEFA

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