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Coordinates, points and lines Distance between two points Using the cartesian coordinate system, the distance between two points A and B is the length of the line segment AB. Using Pythagoras Theorem The distance between the points ( x1 , y1 ) and ( x2 , y2 ) (or the length of the line segment joining them) is x2 x1 y2 y1 2 2 . The mid-point of a line segment The mid-point of a line segment joining x1 , y1 and ( x2 , y2 ) has coordinates 1 1 x x , y y 1 2 1 2 . 2 2 Do Exercise 1A, Q.14, Q.16, Q.20, Q,22, p.7 The gradient of a line segment Unlike distance and the mid-point, the gradient is a property of the whole line, not just of a particular line segment. If you take any two points and find the increase in the x and y coordinates as you go from one to the other, then the fraction y step x step is called the gradient of the line. The gradient of the line joining x1 , y1 to x2 , y2 is y2 y1 . x2 x1 Read Example 1.3.1, 1.3.2, pp.5-6 The equation of a line We can tell whether or not a point is on a line (or curve) by substituting the coordinates into the equation and seeing if the equation is satisfied by the coordinates of the point. The equation of a line or curve is a rule for determining whether or not the point with coordinates x, y lies on the line or curve. To find the equation of the line with gradient m through the point A with coordinates x1 , y1 consider the general point P with coordinates ( x, y ). The gradient of AP is y y1 . x x1 The equation of the line through x1 , y1 with gradient m is y y1 m x x1 . Recognising the equation of a straight line It is easy to see that the line equation y y1 m x x1 can be rewritten in the form y mx c, and that the point 0,c lies on the y-axis. The number c is called the y - intercept of the line. To find the x-intercept, put y 0 in the equation, which gives c x , where m 0. m If m 0, the line has equation y c. Recognising the equation of a straight line Straight lines parallel to the y-axis have equations of the form x k. Note that the line x k does not have a gradient; its gradient is undefined. Its equation cannot be written in the form y mx c. To find the intersection of two (non-parallel) straight lines simply solve the associated simultaneous equations. Do Exercise 1B, Q.5-Q.13, p.12 The gradients of perpendicular lines If the gradient of a line segment PB is m, you can draw a 'gradient triangle' PAB in which PA is one unit and AB is m units. If the gradient triangle PAB is rotated through a right-angle to PAB, then y -step 1 gradient of PB . x-step m Two lines with gradients m1 and m2 are perpendicular if m1m2 1, or m1 1 , m2 or m2 Do Exercise 1C, pp. 14-15 Do Misc. Exercise 1, Q.13-Q.22, p.16 1 . m1