## Elementary Geometry for College Students (6th Edition)

Published by Brooks Cole

# 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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