In a right-angled triangle one of the angles is a right angle, i.e. it measures 90º. The longest side of a right-angled triangle is called the hypotenuse and the other two sides are called the side adjacent to angle A (or C) and the side opposite angle A (or C), in a triangle where B is the right angle.  

Remember that the angles in any triangle add up to 180º. Therefore, in a right-angled triangle the sum of the two acute angles is 90º.

1. In the following window move the vertices of the triangle until you get a right angle at point A. Repeat the process until you construct three triangles that are different in size and shape. In each case write down the value of the angles B and C in your notebook. Check that they add up to 90º. Also, write down the lengths of the three sides of the triangle. 

The ancient Egyptians already knew that in a right-angled triangle with sides 3,4 and 5 units long the following was true. That the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides:

5²=4²+3²

2. In the window above construct a triangle whose sides measure 3,4 and 5. Check that the triangle is a right-angled triangle and that the right angle is the angle opposite the hypotenuse (the side that measures 5). Increase the scale of the window to make this easier to do.

3. Prove that the numbers 10, 8 and 6 (double the values of 5, 4 and 3) have the same relation to each other as the one above. Any multiple of these three numbers 5*k, 4*k and 3*k (where k is any whole number) satisfies the same relation. Change the values of the parameter k in the following graph and prove that the triangles with these measurements are always right-angled triangles and consequently satisfy the relation given earlier.

4. Do the numbers have the same relation when k is a decimal number? Use the following window to check.

Pythagoras was a Greek philosopher and mathematician who lived around the year 500 BC. He discovered that this relation between these numbers is true in any right-angled triangle. So, in a right-angled triangle where the hypotenuse is a and the other two sides b and c the following relation is always true:

a²=b²+c²

This relation is known as PYTHAGORAS' THEOREM

In a right-angled triangle the square of the length of the hypotenuse is the sum of the squares of the lengths of the other two sides.

In the following window use the mouse to move vertex B and construct different right-angled triangles. You can make the scale bigger or smaller at the top of the diagram. You can also change the lengths of sides b and c yourself by using the arrows in the lower part of the window.

5. Notice that the values of the hypotenuse and other two sides (b and c) change when you move vertex B. You will notice that Pythagoras' theorem is true in each case. 

6. Move point B so that b and c have the following values: 8 and 6, 6 and 8, 5 and 12, 12 and 16, 9 and 12, 8 and 15, 20 and 21, and 10 and 10. In each case write down the lengths of the sides of the triangle in your notebook as well as the squares of these measurements (a², b² y c2).

7. Measure the lengths of the sides of your Maths book with a ruler and write them down in your notebook. Call the longest side of the book b and the shortest side c. Put the values of sides b and c into the window above and write down the value given for the hypotenuse a. Measure the diagonal of your book with a ruler and see if it is the same as the value given for a in the window.

8. We have a ladder which is 220cm long and is leaning against a wall 180cm high. How far away from the wall is the base of the ladder?

9. Work out the height of an equilateral triangle whose sides are 6cm long.

The following diagram proves Pythagoras' theorem is true.

In the diagram there are two identical squares whose sides measure b+c. In each square we have positioned four identical right-angled triangles in different places, whose hypotenuse is a and other two sides are b and c.

In the square on the left, once the four triangles have been positioned we get a space which is a square in the middle whose sides are a, equal to the hypotenuse of the triangle. Therefore, the area of this square is a².

In the square on the right there are two spaces that are squares whose sides are b and c. Therefore, their areas are b² and c² respectively.

As the original squares are exactly the same, the spaces in each square also have the same area. In the square on the left,  a² and in the square on the right, b²+c².

Therefore  a² = b²+c²

10. In the following window give sides b and c the values given in exercise 6 and prove that they satisfy Pythagoras' theorem. In each case work out the value of the hypotenuse a.