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Four vehicles travelled on a road. What can you deduce from the times that they met?

Intersections

Change one equation in this pair of simultaneous equations very slightly and there is a big change in the solution. Why?

Graphical Triangle

What is the area of the triangle formed by these three lines?

Perpendicular Lines

Age 14 to 16
Challenge Level

We received correct solutions from Ash and Lucy, two students at Tiffin Girls' School. Well done to you both.
Ash sent us this correct solution:

The gradient of the first line must be the negative reciprocal of the gradient of the other line:
if you multiply the two gradients you always get -1.

For example:

Take a random gradient, say $\frac{4}{7}$
The negative reciprocal gradient will be $\frac{-7}{4}$

Make up two equations with these gradients, say
$y =\frac{4x}{7}$ and $y =\frac{-7x}{4}$

Draw them on a grid

perpendicular lines
You get perpendicular lines.


Lucy recorded how she worked through this problem:

$y = x$ is perpendicular to $y = -x$

$y = x$ is perpendicular to $y = -x + 2$
(the y-intercept doesn't affect the gradient of the line)

$y = 2x$ is perpendicular to $y = -x/2$

$y = -3x$ is perpendicular to $y = x/3 $
(to make a line perpendicular you need to invert the gradient, or take the reciprocal, and change the sign)

$y = -2x$ is perpendicular to $y = x/2 $

I can see a pattern here: when the two gradients of perpendicular lines are multiplied together they give -1, and the y-intercept does not affect if the line is perpendicular or not.

I will now try to work out what the perpendicular line of some other lines will be using this formula:

$y = 7x - 3$:

Using my formula I predict that a line which is perpendicular to this line will be $y = -x/7 - 3$

perpendicular lines
When I tested the lines out, I found that the formula had worked.

$y = x/3 + 4$:

Using my formula I predict that a line which is perpendicular to this line will be $y = -3x + 4$

perpendicular lines
Having drawn out the lines I found that the formula worked and the lines were perpendicular.

$y = -7x/3 + 2$:

I predict that a line which is perpendicular to this line will be $y = 3x/7$

perpendicular lines
From drawing out these lines I can see that they are perpendicular.