Concept explainers
(a)
To check: The postulate that guarantees the existence of the plane ABC, ABD, ACD, BCD .
(a)
Answer to Problem 18WE
It is the postulate 7.
Explanation of Solution
Given information:
Consider the statements below and associated figure
Points A , B , C and D are four non coplanar points
Formula used:
Through any 3 non collinear points there is exactly one plane
Calculation:
From Postulate 7, through any 3 non collinear points there is exactly one plane. From the figure, points A , B , C and D are non collinear points, therefore from above argument there will be exactly one plane through points A,B,C and A,B,D and A,C,D and B,C,D
Hence, the answer is postulate 7.
Conclusion:
It is the postulate 7.
(b)
To describe:The ruler postulate that guarantees the existence of point P between A and D .
(b)
Explanation of Solution
Given information:
Consider the statements below and associated figure
Points A, B, C and D are four non coplanar points
Formula used:
Through any 3 non collinear points there is exactly one plane
Calculation:
By Ruler Postulate, the distance between any two points equals the absolute value of the difference of coordinates. If P is between A and D , with the respective coordinates y, x , and z and assuming x < y < z
By the ruler postulate,
But y-x is positive because x < y. By similar argument, z - y and z - x are positive.
Therefore,
Therefore, the assumption of P is between A and D is correct.
Hence, the answer is P between A and D is proved by ruler postulate.
Conclusion:
The answer is P between A and D is proved by ruler postulate.
(c)
To write:The postulate that guarantees the existence of BCP.
(c)
Answer to Problem 18WE
The answer is postulate 7.
Explanation of Solution
Given information:
Consider the statements below and associated figure
Points A, B, C and D are four non coplanar points.
Formula used:
Through any 3 non collinear points there is exactly one plane
Calculation:
From Postulate 7, through any 3 non collinear points there is exactly one plane. From the figure, points B, C and P are non collinear points therefore from above argument there will be exactly one plane through points B, C, P
Hence, the answer is postulate 7.
Conclusion:
The answer is postulate 7.
(d)
To show:how there are infinite number of planes through
(d)
Answer to Problem 18WE
The infinite number of point on
Explanation of Solution
Given information:
Consider the statements below and associated figure
Points A, B, C and D are four non coplanar points
Formula used:
Through any 3 non collinear points there is exactly one plane
Calculation:
From the definition, a line segment consists of infinite number of points. This implies, there are an infinite number of points on
From Postulate 7, through any 3 non collinear points there is exactly one plane which implies each set of the three points forms a unique plane.
Hence, the answer to is infinite number of point on
Conclusion:
The infinite number of point on
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