OCR Advanced FSMQ - Additional Maths

From the graph on page 118, length AC = x 2 − x 1 and length BC = y 2 − y 1 Gradient of line AB = BC AC = y 2 − y 1 x 2 − x 1 The gradient of the line joining points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by: Gradient = y 2 − y 1 x 2 − x 1 For example, the gradient of the straight line AB joining the points A (−3, 2) and B (1, 6) is 6 − 2 1 − (−3) = 4 4 = 1 6.2 Calculating the distance between two points Suppose you are given the coordinates of two points and are asked to find the distance between them, then it always best to do a quick sketch graph showing their positions. You can then make a right-angled triangle and use the line joining the two points as the hypotenuse of the triangle. There is a formula for working out the distance between two points and the derivation of this formula is shown here. Now AC = x 2 − x 1 and length BC = y 2 − y 1 By Pythagoras’ theorem AB 2 = AC 2 + BC 2 So AB 2 = ( x 2 − x 1 ) 2 + (  y 2 − y 1 ) 2 AB = √ ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 Note that this formula can be expressed in words as ‘the gradient is the difference in the y -coordinates divided by the difference in the x -coordinates. > > >  TIP Be systematic about the way you use this formula. It is a good idea to take the coordinates of the first point in the question to be ( x 1  , y 1 ) and the second point to be ( x 2  , y 2 ). This will make checking your work easier. Do not worry about which point to call ( x 1, y 1 ) or ( x 2 ,  y 2 ). It does not matter, as you will get the same answer whichever you choose. y x C O B ( x 2 , y 2 ) A ( x 1 , y 1 ) > > >  TIP This is a difficult formula to remember. If you forget it, draw a sketch graph showing the line joining the points and form a triangle and work out the lengths and then use Pythagoras' theorem to work out the length of hypotenuse. 6 Coordinate geometry of straight lines 119