Parallel Lines, Perpendicular Lines and Intersections

Aims:

To know how to recognise parallel and perpendicular lines.

To be able to find points of intersection.

To be able to find a perpendicular bisector.

Parallel Lines

Have the same gradient!

To solve problems rearrange straight line equations into y=mx+c and compare the m values.

Testing if Lines are Parallel

Are the lines parallel?

12 3 9 and -8 2 14x y x y

Graphs of Parallel Lines

The red line is the graph of y = – 4x – 3 and the blue line is the graph ofy = – 4x – 7

Practise Testing if Lines are Parallel

Are the lines 6 3 5 and 2 4 4x y y x parallel?

Are the lines 2 4 and 2 4 12x y x y parallel?

Constructing Parallel Lines

Find the equation of a line going through the point (3, -5) and parallel to 2 83y x

Practise Constructing Parallel Lines

Find the equation of the line going through the point (4,1) and parallel to 3 7y x

Find the equation of the line going through the point (-2,7) and parallel to 2 8x y

Perpendicular Lines

Perpendicular lines are lines that intersect at right angles.

You can tell if lines are perpendicular by comparing their gradients.

If one line has a gradient of m the other line must have a gradient of

The gradients of perpendicular lines always multiply to give -1.

m

1

Testing if Lines Are Perpendicular

1Are the lines 2 5 and 4 perpendicular?

2x y y x

Graphs of Perpendicular Lines

The red line is the graph of y = – 2x + 5 and the blue line is the graph ofy = 1/2 x +4

Practise Testing if Lines Are Perpendicular

Are the lines 6 3 5 and 2 4 4 perpendicular?x y y x

Are the lines 2 4 and 4 2 6 perpendicular?x y x y

Constructing Perpendicular Lines

Find the equation of a line going through the point (3, -5) and perpendicular to 2 83y x

Practise Constructing Perpendicular Lines

Find the equation of the line going through the point (4,1) and perpendicular to 3 7y x

Find the equation of the line going through the point (-2,7) and perpendicular to 2 8x y

Perpendicular Bisectors

A perpendicular bisector is a line that is perpendicular to another and also crosses at it’s centre.

To find a perpendicular bisector you must find the midpoint of the line and also the perpendicular gradient.

Perpendicular Bisector Task

You have been given the question and all the workings out to find a perpendicular bisector.

Your task is to explain what is happening in each stage.

Intersecting lines

Intersect means that the lines cross.

To find out where two lines cross you can treat them as simultaneous equations.

The solution to the simultaneous equation is where the lines cross.

Finding the point of intersection

Where do the lines 2y + 5x = 8 and 3y + 2x = 1 intersect?

Practise finding the point of intersection

Where do the lines 5y + x = 13 and 2x – 3y = -5 intersect?

Where do the lines 3y - 5x = 14 and 4y + 2x = 10 intersect?

These lines…..

Ay = 4x + 4

B4y = x + 3

Cy = 8x – 3

Dy + 4x + 6 = 0

E3y = 2x – 8

Fy + 6x = 11

Gy + 8x = 6

H2y + 8 = 3x

I2y + x = 4

J2y = 8x + 3

Ky = 6x – 4

Ly + x + 8 = 0

These lines are parallel:  These lines are perpendicular:  These lines have the same y intercept:  These lines have the same x intercept:  These lines both go through the point (1, 5):  These lines:

Independent Study

Core 1: Exercise 1F - page 23

Mymaths: Equation of a line and Intersecting Lines tasks.

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