In the previous section we completed the discussion on the 'Sum of two sides of a triangle'. In this section, we will discuss about the Pythagoras theorem.
The Pythagoras theorem is related to right angled triangles. A right angled triangle is a triangle in which the interior angle at one corner is a right angle (90o).
In a triangle, there will be only one 90o angle. This is because, if there are two, their sum will be 180o. As the sum of the interior angles of any triangle is 180o, there will not be a third angle, if their are two 90o. And with out 3 angles, a triangle will not be formed.
The fig.9.21 below shows a right angled triangle ABC. The 90o angle is at B.
■ The side opposite to the 90o angle is AC. This side which is opposite to the 90o angle is called the Hypotenuse.
■ The Hypotenuse will be the longest side in a right angled triangle
■ The other two sides AB and BC are called the legs.
Let us do an experiment:
• Make a cardboard cut out of a right angled triangle. It can be of any convenient size.
♦ Mark the hypotenuse as ‘a’.
Mark one of the legs as ‘b’, and the other leg ‘c’. Any of the legs can be marked as ‘b’ or ‘c’. Let us mark
♦ The longer leg as ‘b’ and
♦ The shorter leg as ‘c’.
• Make 7 copies of this right angled triangle. So we have 8 identical right angled triangles as shown in the fig.22(a) below:
• Draw two squares on a sheet of paper. This is shown in fig.9.22(b). The sides of both the squares must be (b + c).
Note that, we made the triangles with any convenient lengths ‘a’, ‘b’ and ‘c’. But there is no such liberty for drawing the squares. The side must be exactly equal to (b + c).
Now take any four of the triangles and arrange them inside the square 1 as shown in fig.9.23(a). Take the remaining four triangles and arrange them inside square 2 as shown in fig.9.23(b)
Now we have to do some calculations based on the above two arrangements:
• Fig 9.23(a): Inside square 1, a smaller square is formed.
♦ Note that all the four sides (of this smaller square) are equal.
♦ Because each one of these four sides is the hypotenuse ‘a’ of our triangle.
♦ So the area of the smaller square is a2.
♦ This a2 is the uncovered area inside square 1.
• Fig 9.23(b): Inside square 2, two smaller squares (the uncovered areas) are formed.
♦ Look at the upper square. It has the sides equal to the smaller leg ‘c’ of our triangle. So this upper square has an area of c2
♦ Look at the lower square. it has the sides equal to the longer leg ‘b’ of our triangle. So this lower square has an area of b2
♦ So the total uncovered area in square 2 is b2 + c2
• Square 1 and square 2 are identical. So they have the same total area. And each of these squares are covered with 4 identical triangles.
• So the uncovered areas in both the squares are the same. Thus we get a2 = b2 + c2
■ This is Pythagoras theorem. It can be stated as:
Square of the hypotenuse = Sum of the squares of the legs.
This property was discovered by Pythagoras, a Greek philosopher who lived in the sixth century BC.
Another experiment: Draw a right angled triangle of any convenient size as shown in the fig.9.24(a)
Now, draw squares on the hypotenuse and the two legs as shown in fig.(b). Compute the areas of each of these squares. We will find that a2 = b2 + c2. So this experiment proves the Pythagoras theorem.
In the next section we will continue our discussion on Pythagoras Theorem.
The Pythagoras theorem is related to right angled triangles. A right angled triangle is a triangle in which the interior angle at one corner is a right angle (90o).
In a triangle, there will be only one 90o angle. This is because, if there are two, their sum will be 180o. As the sum of the interior angles of any triangle is 180o, there will not be a third angle, if their are two 90o. And with out 3 angles, a triangle will not be formed.
The fig.9.21 below shows a right angled triangle ABC. The 90o angle is at B.
 |
| Fig.9.21 |
■ The Hypotenuse will be the longest side in a right angled triangle
■ The other two sides AB and BC are called the legs.
Let us do an experiment:
• Make a cardboard cut out of a right angled triangle. It can be of any convenient size.
♦ Mark the hypotenuse as ‘a’.
Mark one of the legs as ‘b’, and the other leg ‘c’. Any of the legs can be marked as ‘b’ or ‘c’. Let us mark
♦ The longer leg as ‘b’ and
♦ The shorter leg as ‘c’.
• Make 7 copies of this right angled triangle. So we have 8 identical right angled triangles as shown in the fig.22(a) below:
 |
| Fig.9.22 |
• Draw two squares on a sheet of paper. This is shown in fig.9.22(b). The sides of both the squares must be (b + c).
Note that, we made the triangles with any convenient lengths ‘a’, ‘b’ and ‘c’. But there is no such liberty for drawing the squares. The side must be exactly equal to (b + c).
Now take any four of the triangles and arrange them inside the square 1 as shown in fig.9.23(a). Take the remaining four triangles and arrange them inside square 2 as shown in fig.9.23(b)
 |
| Fig.9.23 |
• Fig 9.23(a): Inside square 1, a smaller square is formed.
♦ Note that all the four sides (of this smaller square) are equal.
♦ Because each one of these four sides is the hypotenuse ‘a’ of our triangle.
♦ So the area of the smaller square is a2.
♦ This a2 is the uncovered area inside square 1.
• Fig 9.23(b): Inside square 2, two smaller squares (the uncovered areas) are formed.
♦ Look at the upper square. It has the sides equal to the smaller leg ‘c’ of our triangle. So this upper square has an area of c2
♦ Look at the lower square. it has the sides equal to the longer leg ‘b’ of our triangle. So this lower square has an area of b2
♦ So the total uncovered area in square 2 is b2 + c2
• Square 1 and square 2 are identical. So they have the same total area. And each of these squares are covered with 4 identical triangles.
• So the uncovered areas in both the squares are the same. Thus we get a2 = b2 + c2
■ This is Pythagoras theorem. It can be stated as:
Square of the hypotenuse = Sum of the squares of the legs.
This property was discovered by Pythagoras, a Greek philosopher who lived in the sixth century BC.
Another experiment: Draw a right angled triangle of any convenient size as shown in the fig.9.24(a)
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| Fig.9.24 |
In the next section we will continue our discussion on Pythagoras Theorem.
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