Higher Maths 1 1 Straight Line UNIT OUTCOME SLIDE

UNIT OUTCOME SLIDE NOTE Higher Maths 1 1 Straight Line 2 UNIT OUTCOME SLIDE NOTE Gradient of a Straight Line The symbol is called ‘delta’ and is used to describe change. Example (-2,7) (3,9) x = 3 – (-2) = 5 B ( x2 , y2 ) Gradient y y2 – y1 y2 – y1 m = = x x2 – x1 x2 – x1 A ( x1 , y1 )

UNIT OUTCOME SLIDE NOTE y x y x y = x = x y Higher Maths 1 1 Straight Line 3 UNIT OUTCOME SLIDE NOTE Horizontal and Vertical Lines Horizontal • zero gradient y • parallel to -axis x y = • equations written in the form x Vertical (‘undefined’) • infinite gradient y • parallel to -axis y x = • equations written in the form x

UNIT OUTCOME SLIDE NOTE q y q q x q q m m = tan x y x Higher Maths 1 1 Straight Line 4 UNIT OUTCOME SLIDE NOTE Gradient and Angle The symbol is called ‘theta’ and is often used for angles. q y opp y m tan q = = = x adj opp q x adj m = tan q where is the angle between the line and the positive direction of the -axis. q x

x Collinearity y C A Points are said to be collinear if they lie on the same straight. A C x y O x1 x2 B The coordinates A,B C are collinear since they lie on the same straight line. D,E,F are not collinear they do not lie on the same straight line. D E F

( ) , UNIT OUTCOME SLIDE NOTE x2 x1 y2 y1 + + ( x1 , y1 ) M Higher Maths 1 1 Straight Line 6 UNIT OUTCOME SLIDE NOTE ( x1 , y1 ) Finding the Midpoint The midpoint M of two coordinates is the point halfway in between and can be found as follows: M • find the distance between the x-coordinates • add half of this distance to the first x-coordinate • do the same again for the y-coordinate ( x2 , y2 ) • write down the midpoint An alternative method is to find the average of both coordinates: ( ) x2 x1 y2 y1 + + , M = The Midpoint Formula 2 2

UNIT OUTCOME SLIDE NOTE Higher Maths 1 1 Straight Line 7 UNIT OUTCOME SLIDE NOTE Collinearity If points lie on the same straight line they are said to be collinear. Example Prove that the points P(-6,-5), Q(0,-3) and R(12,1) are collinear. mPQ = (-3) – (-5) 2 1 We have shown = = 6 3 mPQ = mQR 0 – (-6) mQR = 1 – (-3) 4 1 The points are collinear. = = 12 – 0 12 3

UNIT OUTCOME SLIDE NOTE Higher Maths 1 1 Straight Line 8 UNIT OUTCOME SLIDE Q NOTE Gradients of Perpendicular Lines A When two lines are at 90° to each other we say they are perpendicular. If the lines with gradients m1 and m2 are perpendicular, B P m1 = –1 mAB = -3 mPQ = 5 m2 5 3 To find the gradient of a perpendicular line: • flip the fraction upside down • change from positive to negative (or negative to positive)

UNIT OUTCOME SLIDE NOTE Higher Maths 1 1 Straight Line 9 UNIT OUTCOME SLIDE NOTE Different Types of Straight Line Equation Any straight line can be described by an equation. The equation of a straight line can be written in several different ways . Basic Equation (useful for sketching) y = m x + c (not the -intercept!) y General Equation (always equal to zero) A x + B y + C = 0 Straight lines are normally described using the General Equation, with as a positive number. A

UNIT OUTCOME SLIDE NOTE y x y – b x – a y – b = m ( x – a ) Higher Maths 1 1 Straight Line 10 UNIT OUTCOME SLIDE NOTE The Alternative Straight Line Equation Example Find the equation of the line with gradient and passing through (5,-2). y ( x , y ) 1 2 ( a , b ) x Substitute: y – b m = y – (-2) = ( x – 5 ) 1 x – a 2 y + 2 = ( x – 5 ) 1 y – b = m ( x – a ) 2 2y + 4 = x – 5 for the line with gradient m passing through (a,b) x – 2y – 9 = 0

UNIT OUTCOME SLIDE NOTE Higher Maths 1 1 Straight Line 11 UNIT OUTCOME SLIDE NOTE Straight Lines in Triangles An altitude is a perpendicular line between a vertex and the opposite side. = A median is a line between a vertex and the midpoint of the opposite side. A perpendicular bisector is a perpendicular line from the midpoint of a side. = =

UNIT OUTCOME SLIDE NOTE Higher Maths 1 1 Straight Line 12 UNIT OUTCOME SLIDE NOTE A Finding the Equation of an Altitude To find the equation of an altitude: B • Find the gradient of the side it is perpendicular to ( ). mAB C • To find the gradient of the altitude, flip the gradient of AB and change from positive to negative: maltitude = –1 mAB • Substitute the gradient and the point C into Important Write final equation in the form y – b = m ( x – a ) A x + B y + C = 0 with A x positive.

( ) UNIT OUTCOME SLIDE NOTE x2 x1 + y2 y1 + , y – b = m ( x – a ) = = Higher Maths 1 1 Straight Line 13 UNIT OUTCOME SLIDE NOTE Q Finding the Equation of a Median M = To find the equation of a median: = P • Find the midpoint of the side it bisects, i.e. x2 x1 + y2 y1 + O ( ) , M = 2 2 • Calculate the gradient of the median OM. Important • Substitute the gradient and either point on the line (O or M) into Write answer in the form A x + B y + C = 0 y – b = m ( x – a ) with A x positive.