Any two points on a grid which are joined together with a straight line make a segment. We can work out the length of this segment easily by using the basic unit of measurement which is the distance between two consecutive points on the grid. Think of the segment as the hypotenuse of a right-angled triangle. The two shorter sides of the triangle are easy to determine. 

Play around with different sized segments by moving the points using the cursors or arrows.

In the first example the shorter sides are 3 and 2 units long respectively. This allows us to get a measurement of for the segment although in the diagram the side is given the approximate length of 3.61.

Exercise 1

You need to use a systematic method to be able to find out all the possible measurements of segments on the grid. The following table can help you to do this and you should copy and complete it into your notebook.

Length of the other two sides	(1,0)	(2,0)	(3,0)	(4,0)	(5,0)	(1,1)	(2,1)	(3,1)	...
Length of the segment	1	2	3	4	5		...	...	...

Exercise 2

Use a table like the one above to help you write all the possible lengths of segments on a 6x6 grid.

The idea is to form as many squares as possible on a 5x5 grid making sure that the corners of the squares coincide with the points on the grid. Using points A and B by way of an example, position points C and D so that ABCD is a square. 

Use the cursors to move points A and B but make sure that the segment is the same length as before. Then, move points C and D until you make another square on the grid.

Exercise 3

Find out how many squares you can construct on a 5x5 grid whose sides are the same measurements as those in the previous activity.

You need to organize the information properly so that you don't miss any of the squares. Use the table above again and add and complete two new rows:

Length of the other two sides	(1,0)	(2,0)	(3,0)	(4,0)	(5,0)	(1,1)	(2,1)	(3,1)	...
Length of the segment	1	2	3	4	5		...	...	...
Area of the square	1	4	...	...	...	...	...	...	...
Number of squares	16	9	...	...	...	...	...	...	...

Exercise 2

Use a table like the one above to help you write all the possible lengths of segments on a 6x6 grid in your notebook.

Using points A and B we have to try and locate point P on the grid so that the three points form a right-angled triangle. Find all the possible positions for P without moving points A and B. If two triangles can be superimposed then they have the same solution. If you are in any doubt use the method indicated in activity I to measure the sides of the triangle and check it by applying Pythagoras' theorem.

Exercise 4

Move points A and B to different positions and work out all the possible solutions for P in each case.

Exercise 5

Are there any pairs of A and B points that can't be used to make a right-angled triangle with point P?

In the triangle in the diagram we can take side AB to be the base unit of measurement of 1. Its height is also 1. However, different triangles with a height of 1 can be formed by moving point P along the line parallel to the base. Move the point P to see four more triangles with a base and height of 1.

Exercise 6

Make the base AB = 2 and move point P to find different triangles with a height of 2.

Exercise 7

Find out all the possible measurements until you get all 22 solutions. Once again you should record the information in a table in your notebook.