Elementary Geometry for College Students (6th Edition)

Published by Brooks Cole
ISBN 10:
ISBN 13:

Chapter 10 - Test - Page 488: 23

Answer

(a) The sphere has symmetry with respect to the xy plane. (b) The sphere has symmetry with respect to this line.

Work Step by Step

We can write the general equation of a sphere: $(x-a)^2+(y-b)^2+(z-c)^2 = r^2$ where $(a,b,c)$ is the center of the sphere and $r$ is the radius The equation of the sphere is: $x^2+y^2+z^2 = 25$ The center of the sphere is $(0,0,0)$ (a) A sphere has symmetry with respect to any plane that includes the sphere's center. Since the xy plane includes the point $(0,0,0)$, the sphere has symmetry with respect to the xy plane. (b) When $n = 1$, the line $(-1,2,3)+n(1,-2,-3)$ passes through the point $(0,0,0)$ A sphere has symmetry with respect to any line that passes through the sphere's center. Since this line passes through the point $(0,0,0)$ which is the sphere's center, the sphere has symmetry with respect to this line.
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