Linear Geometry

You have already completed the bridging booklet Reminder You have already completed the bridging booklet chapter 9 Linear graphs If you need support with any of the following you MUST see your teacher Rearranging a linear equation to find the gradient and intercept Drawing a line from an equation

Starter: Ten questions Linear coord geometry KUS objectives BAT explore gradients of parallel and perpendicular line BAT rearrange and find equations of lines Starter: Ten questions Identify the equation of each of the following line graphs

Starter: identify linear equations x Starter: identify linear equations Identify the equations of these lines

Answers y = 2x + 3 y = 2x - 4 y = 3x + 1 y = -2x + 5 y = x + 2 y = 3

Notes: General equation of a line We are used to: gradient y-intercept Another ‘standard way to write the equation of a line is: The General form of the equation of a line

For each of these equations, i) rearrange it into the form y = mx + c WB1 gradient and y intercept For each of these equations, i) rearrange it into the form y = mx + c ii) give the gradient iii) give the intercept on the y-axis. 10 Gradient m = -2 Gradient m = 2.5 3 Intercept (0, 10) Intercept (0, 3)

y y = 3 x x y = 3 x + 1 y = 3 x + 2 y = 3 x + 3 y = 3 x + 4 Notes 1 y = 3 x y = 3 x + 1 y = 3 x + 2 y = 3 x + 3 y = 3 x + 4 y = 3 x + 5 y = 3 x – 1 y = 3 x – 2 x y = 3 x – 3 y = 3 x – 4 The ‘family’ of Parallel lines with equation y = 3x + a

There are three sets of parallel lines here. Challenge 1 There are three sets of parallel lines here. Match them up and say what the gradient is for each set.

General equation intersection points Why bother with the general equation? Solve these simultaneous equations: Easier to work with:

General equation intersection points Why bother with the general equation? Solve these simultaneous equations: The solution is the Intersection point of the two lines (-2, -½)

Which of these lines are parallel to the line 3x - 2y – 4 = 0 Challenge 2 Which of these lines are parallel to the line 3x - 2y – 4 = 0

These lines are perpendicular Notes These lines are perpendicular Define perpendicular = at right angles

Notes What is each gradient? 1 2 Gradient = ½ 2 Gradient = -2 1

Notes What is each gradient? Gradient = 3 1 3 3 1 Gradient = -1/3

PERPENDICULAR LINE IS THE Notes 1 THE GRADIENT OF A PERPENDICULAR LINE IS THE NEGATIVE RECIPROCAL OF THE OTHER 𝒎 𝟏 × 𝒎 𝟐 =−𝟏

y x What do you notice? Draw a Perpendicular line to y = 3x Notes Draw a Perpendicular line to y = 3x What do you notice? The ‘family’ of Perpendicular lines with general equation y = -1/3x + a

gradient y-intercept 2 -5 Examples: We are used to: Notes We are used to: gradient y-intercept The Perpendicular line has gradient m2 where: line perpendicular 2 -5 Examples:

What is the gradient of the lines perpendicular to these? Practice 1 What is the gradient of the lines perpendicular to these? y = 2x + 1 m = 2 -1/m= -1/2 y = 2 + 4x m = 4 -1/m= -1/4 y = 3x + 2 m = 3 -1/m= -1/3 y + 2x = 2 m = -2 -1/m= 1/2 2y = 3x - 2 m = 3/2 -1/m= -2/3 5y + 2x = 3 m = -2/5 -1/m= 5/2

WB2 Give the General equation of the perpendicular line to 2𝑥+𝑦−8=0 that goes through (4, 9) Has gradient m1 = -2 So the gradient of a perpendicular line is m2 = ½ When x = 4, y = 9 … so

WB3 Give the General equation of the perpendicular line to 𝑥+5𝑦−6=0 that goes through ( 3 5 , 7) Has gradient m1 = -1/5 So the gradient of a perpendicular line is m2 = 5 When x = 3/5, y = 7 … so

First the midpoint of line AB is (−1. 4) WB4 Two points A(1,2) and B(-3,6) are joined to make the line AB. Find the equation of the perpendicular bisector of AB First the midpoint of line AB is (−1. 4) 𝑚 1 = 6−2 −3−1 =−1 2nd the gradient of line AB is So the perpendicular gradient is 𝑚 2 =1 𝑦=𝑥+ ? When x =1, y = 2 so 𝑦=𝑥+1 Is the perpendicular bisector of line AB

BAT rearrange and find equations of lines KUS objectives BAT explore gradients of parallel and perpendicular line BAT rearrange and find equations of lines self-assess One thing learned is – One thing to improve is –

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