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1 Higher Mathematics CfE Edition This document w as produced specially for the website, and we require that any copies or derivative works attribute the work to Higher Still notes . For more details about the copyright on these notes , please see hsn . Straight Lines Contents Straight Lines 1 1 The Distance Between Points A 1 2 The Midpoint Formula A 3 3 Gradients A 4 4 Collinearity A 6 5 Gradients of perpendicular Lines A 7 6 The Equation of a Straight line A 8 7 Medians A 11 8 Altitudes A 12 9 perpendicular Bisectors A 13 10 Intersection of Lines A 14 11 Concurrency A 17 Higher Mathematics Straight Lines Page 1 CfE Edition hsn . Straight Lines 1 The Distance Between Points A Points on Horizontal or Vertical Lines It is relatively straightforward to work out the distance between two points which lie on a line parallel to the x- or y-axis.

2 In the diagram to the left, the points () 11,xy and () 22,xy lie on a line parallel to the x-axis, 12yy=. The distance between the points is simply the difference in the x-coordinates, = 21dx x where 21xx>. In the diagram to the left, the points () 11,xy and () 22,xy lie on a line parallel to the y-axis, 12xx=. The distance between the points is simply the difference in the y-coordinates, = 21dy y where 21yy>. EXAMPLE 1. Calculate the distance between the points () 7,3 and () 16,3 . ( )The distance is 16716 723 units = +=. x () 11,xy() 22,xyyO dx () 11,xy() 22,xyO ydHigher Mathematics Straight Lines Page 2 CfE Edition hsn . The Distance Formula The distance formula gives us a method for working out the length of the Straight line between any two points. It is based on Pythagoras s Theorem.

3 The distance d between the points () 11,xy and () 22,xy is () ()222121d xxyy= + units. EXAMPLES 2. A is the point () 2, 4 and ( ) B 3, 1. Calculate the length of the line AB. () ()( )()()()2221212222 The length is 3 2145325 934 unitsxxyy + = + = + = +=. 3. Calculate the distance between the points () 15124, and () 1,1 . () ()()()()()( )( )222121222222151241521422443112491211643 61211616157161574 The distance is 11 unitsxxyy + = + += + += += += +==. yOx() 22,xy() 11,xy21xx 21yy dNote The 21yy and 21xx come from the method above. Note You need to become confident working with fractions and surds so practise! Higher Mathematics Straight Lines Page 3 CfE Edition hsn . 2 The Midpoint Formula A The point half-way between two points is called their midpoint.

4 It is calculated as follows. The midpoint of () 11,xy and () 22,xy is ++ 1 21 2,22xxy y. It may be helpful to think of the midpoint as the average of two points. EXAMPLES 1. Calculate the midpoint of the points () 1,4 and ( ) 7, 8. ( )( ) 1 21 2 The midpoint is ,228471,224, 2xxy y++ + + = =. 2. In the diagram below, () A 9,2 lies on the circumference of the circle with centre () C 17, 12, and the line AB is the diameter of the circle. Find the coordinates of B. Since C is the centre of the circle and AB is the diameter, C is the midpoint of AB. Using the midpoint formula, we have: ()( ) 9217, 12,where B is the point ,22xyxy+ + =.

5 By comparing x- and y-coordinates, we have: 917293425xxx+=+== += + ==2and So B is the point () 25, 26. ABCNote Simply writing The midpoint is (4, 2) would be acceptable in an exam. Higher Mathematics Straight Lines Page 4 CfE Edition hsn . 3 Gradients A Consider a Straight line passing through the points () 11,xy and () 22,xy: The gradient m of the line through () 11,xy and () 22,xy is 211221change in vertical height for change in horizontal distanceyymxxxx == . Also, since 2121 OppositetanAdjacentyyxx == we obtain: =tanm where is the angle between the line and the positive direction of the x-axis. Note As a result of the above definitions: Lines with positive gradients slope up, from left to right; Lines with negative gradients slope down, from left to right; Lines parallel to the x-axis have a gradient of zero; Lines parallel to the y-axis have an undefined gradient.

6 We may also use the fact that: Two distinct Lines are said to be parallel when they have the same gradient (or when both Lines are vertical). yOx() 22,xy() 11,xy21xx 21yy positive direction xNote is the Greek letter theta . It is often used to stand for an angle. Higher Mathematics Straight Lines Page 5 CfE Edition hsn . EXAMPLES 1. Calculate the gradient of the Straight line shown in the diagram below. tantan 320 62 (to 2 )m == =. 2. Find the angle that the line joining () P2,2 and ( ) Q 1, 7 makes with the positive direction of the x-axis. The line has gradient 212172312yymxx +=== +. And so ( ) ==== 1tantan3tan371 57 (to 2 ).m 3. Find the size of angle shown in the diagram below. We need to be careful because the in the question is not the in tanm = . So we work out the angle a and use this to find : ( )( )11tantan578 690am ===.

7 So 9078 69011 31 (to 2 ) = = . 5m=x O ya 5m=x O y32 x y O Higher Mathematics Straight Lines Page 6 CfE Edition hsn . 4 Collinearity A Points which lie on the same Straight line are said to be collinear. To test if three points A, B and C are collinear we can: 1. Work out ABm. 2. Work out BCm (or ACm). 3. If the gradients from 1. and 2. are the same then A, B and C are collinear. If the gradients are different then the points are not collinear. This test for collinearity can only be used in two dimensions. EXAMPLES 1. Show that the points () P6,1 , ( ) Q 0, 2 and () R 8, 6 are collinear. ( )( )PQ31262106m == = QR41826280m == = Since PQQRmm= and Q is a common point, P, Q and R are collinear. 2. The points ( )A 1, 1 , ()B1,k and ( )C 5, 7 are collinear. Find the value of k.

8 Since the points are collinear ABACmm=: ( )()()17 11151182412 = += + = = ABBCmm so A, B and C are not collinear. ABCABmBCmABBCmm= so A, B and C are collinear. ABCABmBCmHigher Mathematics Straight Lines Page 7 CfE Edition hsn . 5 Gradients of perpendicular Lines A Two Lines at right-angles to each other are said to be perpendicular . If perpendicular Lines have gradients m and m then 1mm = . Conversely, if = 1mm then the Lines are perpendicular . The simple rule is: if you know the gradient of one of the Lines , then the gradient of the other is calculated by inverting the gradient ( flipping the fraction) and changing the sign. For example: == 3232if then .mm Note that this rule cannot be used if the line is parallel to the x- or y-axis. If a line is parallel to the x-axis (0)m=, then the perpendicular line is parallel to the y-axis it has an undefined gradient.

9 If a line is parallel to the y-axis then the perpendicular line is parallel to the x-axis it has a gradient of zero. EXAMPLES 1. Given that T is the point () 1,2 and S is () 4, 5 , find the gradient of a line perpendicular to ST. () == ST755241m So =57m since = ST1mm. 2. Triangle MOP has vertices () M3, 9 , ( ) O 0, 0 and () P 12, 4. Show that the triangle is right-angled. Sketch: OM90303m = = MP51513943 12m = = = OP134012 0m = = Since OMOP1mm = , OM is perpendicular to OP which means MOP is right-angled at O. Note The converse of Pythagoras s Theorem could also be used here: () P 12, 4( ) O 0, 0() M3, 9 Higher Mathematics Straight Lines Page 8 CfE Edition hsn . ( )222OP222OM1241603990dd= +== += ( )()()( )222MP221234 + = + = Since +=22 2 OPOMMPdd d, triangle MOP is right-angled at O.

10 6 The Equation of a Straight line A To work out the equation of a Straight line , we need to know two things: the gradient of the line , and a point which lies on the line . The Straight line through the point ( ) ,ab with gradient m has the equation ().y b mx a = Notice that if we have a point ( ) 0,c the y-axis intercept then the equation becomes y mx c= +. You should already be familiar with this form. It is good practice to rearrange the equation of a Straight line into the form 0ax by c+ += where a is positive. This is known as the general form of the equation of a Straight line . Lines Parallel to Axes If a line is parallel to the x-axis ( 0m=), its equation is yc=. If a line is parallel to the y-axis ( m is undefined), its equation is xk=. x O yk xk=x O yc yc=Higher Mathematics Straight Lines Page 9 CfE Edition hsn.