OCR Advanced FSMQ - Additional Maths

The length of a straight line joining the two points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by: √ ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 For example, the length of the straight line joining the points A  (−3, −1) and B  (1, 2) is √ ( 1 − (−3) ) 2 + ( 2 − (−1) ) 2 = √ 16 + 9 = √ 25 = 5 units Example 1 Find the length of the line joining the two points (−1, −2) and (4, 10). Answer 1 Using the formula for the distance between two points The length of a straight line joining the two points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by: √ ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 Putting the coordinates (−1, −2) and (4, 10) into this gives Length = √ (4 − (−1)) 2 + (10 − (−2)) 2 = √ 25 + 144 = √ 169 = 13 6.3 Proving two lines are the same length using vectors Suppose we want to express A to B as a vector where A is (1, 2) and B is (5, 4). If you look at the following graph you can see that to go from A to B you go 4 units in the positive x -direction and 2 units up in the positive y -direction. This can be written in vector form (i.e. ( x y ) ) as ( 4 2 ) . You can spot this vector without drawing a graph by looking at the increases in the x -coordinates and y -coordinates when you go from A to B . The x -coordinate goes from 1 to 5 (i.e. an increase of 4) and the y -coordinate goes from 2 to 4 (i.e. an increase in 2). The vector can thus be written as ( 4 2 ) . TAKE NOTE Be careful putting negative numbers into this formula. It is best to add brackets to emphasise the negative numbers. Always remember that when a minus number is squared, the result is always positive. > > >  TIP This formula is just an application of Pythagoras’ theorem. 2 Coordinate geometry 120