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5c1. Lines, Angels, Co-ordinate Geometry

The Basic Concepts of lines and Angles


Two points in space define a line 



Only one line pass through these 2 points. A line has infinite number of points.


An angle is the union of two rays with a common end point. The sides of the angles are the two rays, while the vertex is their common endpoint. An angle is represented by Ð and three letters. The first and third letters represent points on each of the rays that form one of the sides. The middle letter represents the vertex. The first and last letters are interchangeable. ÐABC and ÐCBA represent the same angle. Since B is the vertex, it is always in the middle of the two letters.


Co-ordinate Geometry


Coordinates are pairs of numbers that are used to determine points in a plane, relative to a special point called the origin. The origin has coordinates (0, 0).


The two axes X’OX and Y’OY divide the plane into four regions called the quadrants. The ray OX is taken as positive x-axis, OX’ as negative x axis, OY as positive y-axis and OY’ as negative y-axis. The following signs of abscissa and ordinate thus characterize the quadrants:



I. Quadrant x > 0, y > 0 or (+,+) II. Quadrant x < 0, y > 0 or (–,+)


III. Quadrant x < 0, y < 0 or (–,–) IV. Quadrant x > 0, y < 0 or (+,–)


Distance Formula


Let P(x1, y1) and Q(x2, y2) be any two points on the plane, then the distance between P and Q is given by



Relationship between three points (Section Formula)


If P is a point dividing the joint of two points A(x1, y1) and B(x2, y2) internally in the ratio m : n (i.e. PA : PB = m : n) then the co-ordinates (x, y) of P are given by




If P(x, y) divides the joint of A(x1, y1) and B(x2, y2) externally in the ratio m : n (i.e. PA : PB = m : n) then the co-ordinates (x, y) of P are given by




If P is the midpoint of AB, then m = n, and the ratio is 1 : 1. The coordinates of the midpoint of the line segment joining A(x1, y1) and B(x2, y2) is










 
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