ssrs barcode font 4.2B Distance Principles in Objective-C

Generate QR-Code in Objective-C 4.2B Distance Principles

4.2B Distance Principles
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PRINCIPLE 1:
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If a point is on the perpendicular bisector of a line segment, then it is equidistant from the ends of the line segment.
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Thus if P is on CD, the ' bisector of AB in Fig. 4-24, then PA > PB.
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Fig. 4-24
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CHAPTER 4 Parallel Lines, Distances, and Angle Sums
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2: If a point is equidistant from the ends of a line segment, then it is on the perpendicular bisector of the line segment. (Principle 2 is the converse of Principle 1.)
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Thus if PA > PB in Fig. 4-24, then P is on CD, the ' bisector of AB.
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If a point is on the bisector of an angle, then it is equidistant from the sides of the angle.
Thus if P is on AB, the bisector of j A in Fig. 4-25, then PQ > PR, where PQ and PR are the distances of P from the sides of the angle.
Fig. 4-25
PRINCIPLE 4:
If a point is equidistant from the sides of an angle, then it is on the bisector of the angle. (Principle 4 is the converse of Principle 3.)
Thus if PQ PR, where PQ and PR are the distances of P from the sides of j A in Fig. 4-25, then P is on AB, the bisector of jA.
PRINCIPLE 5:
Two points each equidistant from the ends of a line segment determine the perpendicular bisector of the line segment. (The line joining the vertices of two isosceles triangles having a common base is the perpendicular bisector of the base.)
Thus if PA > PB and QA > QB in Fig. 4-26, then P and Q determine CD, the ' bisector of AB.
Fig. 4-26
PRINCIPLE 6:
The perpendicular bisectors of the sides of a triangle meet in a point which is equidistant from the vertices of the triangle.
Thus if P is the intersection of the ' bisectors of the sides of ^ ABC in Fig. 4-27, then PA > PB > PC. P is the center of the circumscribed circle and is called the circumcenter of ^ABC.
Fig. 4-27
CHAPTER 4 Parallel Lines, Distances, and Angle Sums
PRINCIPLE
The bisectors of the angles of a triangle meet in a point which is equidistant from the sides of the triangle.
Thus if Q is the intersection of the bisectors of the angles of ^ ABC in Fig. 4-28, then QR > QS > QT, where these are the distances from Q to the sides of ^ABC. Q is the center of the inscribed circle and is called the incenter of ^ABC.
Fig. 4-28
SOLVED PROBLEMS
Finding distances In each of the following, find the distance and indicate the kind of distance involved. In4 4-29(a), Fig. 4 4 4 find the distance (a) from P to A; (b) from P to CD; (c) from A to BC; (d ) from AB to CD. In Fig. 4-29(b), find the distance (e) from P to inner circle O; (f) from P to outer circle O; (g) between the concentric circles.
Fig. 4-29
Solutions
(a) PA (b) PG (c) AE (d) FG (e) PQ (f) PR (g) QR 7, distance between two points 4, distance from a point to a line 10, distance from a point to a line 6, distance between two parallel lines 12 3 12 5 5 3 9, distance from a point to a circle 7, distance from a point to a circle 2, distance between two concentric circles
Locating a point satisfying given conditions In Fig. 4-30. (a) Locate P, a point on BC and equidistant from A and C. (b) Locate Q, a point on AB and equidistant from BC and AC. (c) Locate R, the center of the circle circumscribed about ^ ABC. (d) Locate S, the center of the circle inscribed in ^ABC.
Fig. 4-30
CHAPTER 4 Parallel Lines, Distances, and Angle Sums
Solutions
See Fig. 4-31.
Fig. 4-31
Applying principles 2 and 4 For each ^ABC in Fig. 4-32, describe P, Q, and R as equidistant points, and locate each on a bisector.
Fig. 4-32
Solutions
(a) Since P is equidistant from A and B, it is on the ' bisector of AB. Since Q is equidistant from B and C, it is on the ' bisector of BC. Since R is equidistant from A, B, and C, it is on the ' bisectors of AB, BC, and AC. (b) Since P is equidistant from AB and BC, it is on the bisector of jB. Since Q is equidistant from AC and BC , 4 4 4 it is on the bisector of jC. Since R is equidistant from AB , BC, and AC, it is on the bisectors of jA, j B, and jC.
4 4 4 4
Applying principles 1, 3, 6, and 7 For each ^ABC in Fig. 4-33, describe P, Q, and R as equidistant points. Also, describe R as the center of a circle.
Fig. 4-33
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