Fundamental Rules of Geometry for Physics Help (page 2)
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.
Suppose that L is a line segment connecting two points P and R . Then there is one and only one perpendicular line M that intersects line segment L in a point Q such that the distance from P to Q is equal to the distance from Q to R . That is, every line segment has exactly one perpendicular bisector. This is illustrated in Fig. 4-6.
Distance Addition And Subtraction
Let P, Q , and R be points on a line L such that Q is between P and R . Then the following equations hold concerning distances as measured along L (Fig. 4-7):
PQ + QR = PR
PR − PQ = QR
PR − QR = PQ
Angle Addition And Subtraction
Let P, Q, R , and S be four points that lie in a common plane. Let Q be the vertex of three angles ∠PQR, ∠PQS, and ∠SQR, as shown in Fig. 4-8. Then the following equations hold concerning the angular measures:
∠PQS + ∠SQR = ∠PQR
∠PQR - ∠PQS = ∠SQR
∠PQR - ∠SQR = ∠PQS
Suppose that L and M are two lines that intersect at a point P . Opposing pairs of angles, denoted x and y in Fig. 4-9, are known as vertical angles and always have equal measure.
Alternate Interior Angles
Suppose that L and M are parallel lines. Let N be a line that intersects L and M at points P and Q , respectively. In Fig. 4-10, angles labeled x are alternate interior angles; the same holds true for angles labeled y . Alternate interior angles always have equal measure. Line N is perpendicular to lines L and M if and only if x = y .
Fig. 4-10 Alternate interior angles have equal measure.
Alternate Exterior Angles
Suppose that L and M are parallel lines. Let N be a line that intersects L and M at points P and Q , respectively. In Fig. 4-11, angles labeled x are alternate exterior angles; the same holds true for angles labeled y . Alternate exterior angles always have equal measure. The line N is perpendicular to lines L and M if and only if x = y .
Let L and M be parallel lines. Let N be a transversal line that intersects L and M at points P and Q , respectively. In Fig. 4-12, angles labeled w are corresponding angles; the same holds true for angles labeled x, y , and z . Corresponding angles always have equal measure. The line N is perpendicular to lines L and M if and only if w = x = y = z = 90° = π/2 radians; that is, if and only if all four angles are right angles.
Suppose that L is a line and P is a point not on L . There exists one and only one line M through P such that line M is parallel to line L (Fig. 4-13). This
is one of the most important postulates in euclidean geometry. Its negation can take two forms: Either there is no such line M , or there exists more than one such line M 1 , M 2 , M 3 ,.... Either form of the negation of this principle constitutes a cornerstone of noneuclidean geometry that is important to physicists and astronomers interested in the theories of general relativity and cosmology.
Let L and M be lines that lie in the same plane. Suppose that both L and M intersect a third line N and that both L and M are perpendicular to N . Then lines L and M are parallel to each other (Fig. 4-14).
Practice problems for these concepts can be found at: Basics Of Geometry for Physics Practice Test
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