Fundamental Rules of Geometry for Physics Help
The fundamental rules of geometry are used widely in physics and engineering. These go all the way back to the time of the ancient Egyptians and Greeks, who used geometry to calculate the diameter of the earth and the distance to the moon. They employed the laws of euclidean geometry (named after Euclid, a Greek mathematician who lived thousands of years ago). However much or little Euclid actually had to do with these rules, they’re straightforward. So here they are, terse and plain.
Suppose that P and Q are two distinct geometric points. Then the following statements hold true, as shown in Fig. 4-1:
- P and Q lie on a common line L .
- L is the only line on which both points lie.
Let P, Q , and R be three distinct points, not all of which lie on a straight line. Then the following statements hold true:
- P, Q , and R all lie in a common euclidean plane S .
- S is the only euclidean plane in which all three points lie.
Principle Of N Points
Let P 1 , P 2 , P 3 ,..., and P n be n distinct points, not all of which lie in the same euclidean space of n − 1 dimensions. Then the following statements hold true:
- P 1 , P 2 , P 3 ,..., and P n all lie in a common euclidean space U of n dimensions.
- U is the only n -dimensional euclidean space in which all n points lie.
The distance between any two points P and Q , as measured from P toward Q along the straight line connecting them, is symbolized by writing PQ .
Suppose that there is a line segment connecting two points P and R . Then there is one and only one point Q on the line segment between P and R such that PQ = QR . This is illustrated in Fig. 4-2.
Imagine that P, Q , and R are three distinct points. Let L be the line segment connecting P and Q ; let M be the line segment connecting R and Q . Then the angle between L and M , as measured at point Q in the plane defined by the three points, can be written as ∠PQR or as ∠RQP . If the rotational sense of measurement is specified, then ∠PQR indicates the angle as measured from L to M , and ∠RQP indicates the angle as measured from M to L (Fig. 4-3.) These notations also can stand for the measures of angles, expressed either in degrees or in radians.
Suppose that there is an angle ∠PQR measuring less than 180° and defined by three points P, Q , and R , as shown in Fig. 4-4. Then there is exactly one ray M that bisects the angle ∠PQR . If S is any point on M other than the point Q , then ∠PQS = ∠SQR. Every angle has one and only one ray that divides the angle in half.
Suppose that L is a line through points P and Q . Let R be a point not on L . Then there is one and only one line M through point R intersecting line L at some point S such that M is perpendicular to L . This is shown in Fig. 4-5.
Fig. 4-5 . Perpendicular principle.
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