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

Published by Brooks Cole

# Chapter 10 - Section 10.6 - The Three-Dimensional Coordinate System - Exercises - Page 483: 42

#### Answer

(a) The sphere has symmetry with respect to this line. (b) The sphere does not have symmetry with respect to this plane.

#### 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: $x^2+y^2+z^2 = 100$ The center of the sphere is $(0,0,0)$ (a) A sphere has symmetry with respect to any line that passes through the sphere's center. Since the line passing through the points $(0,0,0)$ and $(0,5,5\sqrt{5})$ passes through the point $(0,0,0)$, the sphere has symmetry with respect to this line. (b) A sphere has symmetry with respect to any plane that includes the sphere's center. Since the plane with the equation $y=5$ does not include the point $(0,0,0)$, the sphere does not have symmetry with respect to this plane.

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