Hello, and welcome to today's mathematics lesson! Today, we are going to explore one of the most elegant and powerful results in geometry — the Pythagoras Theorem. By the end of this lesson, you will understand what this theorem states, how to apply it to find unknown sides in right-angled triangles, and how to use its converse to test whether a triangle is right-angled.
Let us begin with the basics. In any right-angled triangle, the side opposite the 90 degree angle has a special name — it is called the hypotenuse. This is always the longest side of the triangle.
Imagine a triangle ABC where angle ABC equals 90 degrees. The side opposite this right angle is AC — and that is your hypotenuse. It stretches across from one leg to the other, forming the diagonal side.
This remarkable relationship was discovered independently by mathematicians across different cultures. An Indian mathematician named Baudhāyana explored this idea, and later the Greek mathematician Pythagoras developed it further. Today, we honour this discovery as the Pythagoras Theorem.
Here is the precise statement of the theorem.
In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Let me say that again, because it is fundamental. The square of the hypotenuse equals the sum of the squares of the remaining two sides.
Consider triangle ABC with angle BAC equal to 90 degrees. Here, BC is the hypotenuse — the side opposite the right angle. The other two sides, AB and AC, are called the legs or the base and perpendicular.
According to the theorem: BC² = AB² + AC²
Notice how the hypotenuse always stands alone on one side of the equation. If you were looking at a different right-angled triangle, say PQR with the right angle at Q, then PR would be your hypotenuse, and you would write: PR² = PQ² + QR²
Let us see this in action with a worked example.
Suppose triangle ABC is right-angled at vertex A. We are given that AB equals 8 centimetres and AC equals 6 centimetres. We need to find the length of BC.
Since the right angle is at A, the side opposite to it — that is BC — must be the hypotenuse. Applying the Pythagoras Theorem: BC² = AB² + AC²
Substituting the values: BC² = 8² + 6², which gives us 64 plus 36, equalling 100.
Therefore, BC equals the square root of 100, which is 10 centimetres.
Here is another situation. Triangle XYZ is right-angled at vertex Z. We know XY equals 15 centimetres and XZ equals 9 centimetres. We need to find YZ.
Since angle Z is 90 degrees, the side opposite to it — XY — is the hypotenuse. So we write: XY² = XZ² + YZ²
Substituting: 225 equals 81 plus YZ². Rearranging, YZ² equals 225 minus 81, which is 144. Thus, YZ equals 12 centimetres.
One more example to build your confidence. Triangle PQR is right-angled at R. Given PQ equals 25 centimetres and QR equals 20 centimetres, find PR.
Here, PQ is the hypotenuse. So PR² + QR² = PQ². This becomes PR² + 20² = 25². So PR² equals 625 minus 400, giving 225. Hence, PR equals 15 centimetres.
Now, let us turn to something equally powerful — the converse of the Pythagoras Theorem.
The converse helps us work backwards. Instead of starting with a right-angled triangle, we start with any triangle and ask: is this right-angled?
Here is the precise statement.
In a triangle, if the square of one side — specifically the longest side — is equal to the sum of the squares of the remaining two sides, then the angle opposite to that longest side is a right angle. Consequently, the triangle is right-angled.
Imagine a triangle where AB is the largest side. If AB² = AC² + BC², then the angle opposite AB — which is angle C — must equal 90 degrees.
Let us test this with an example. The sides of a triangle are 20 centimetres, 9 centimetres, and 12 centimetres. Is this a right-angled triangle?
First, identify the longest side: 20 centimetres. Check whether 20² = 9² + 12². Now, 20 squared is 400. And 9 squared plus 12 squared equals 81 plus 144, which is 225.
Since 400 does not equal 225, the condition fails. Therefore, this triangle is not right-angled.
Here is a beautiful extension that connects to triangle classification.
Consider any triangle ABC where AB is the longest side. There are three possibilities.
First, if AB² = BC² + AC², then angle C equals 90 degrees — a right-angled triangle.
Second, if AB² > BC² + AC², then angle C is greater than 90 degrees — an obtuse-angled triangle.
Third, if AB² < BC² + AC², then angle C is less than 90 degrees — an acute-angled triangle.
This gives you a complete tool for classifying triangles just by comparing squares of sides.
Let us work through one more practical example involving two right-angled triangles sharing a common side.
Suppose angle ACB and angle ACD are both 90 degrees. We are given AB equals 25 centimetres, AD equals 17 centimetres, and AC equals 15 centimetres. We need to find BC, CD, and finally BD.
In triangle ABC, applying the Pythagoras Theorem: AC² + BC² = AB² So 225 plus BC² equals 625. This gives BC² equals 400, so BC equals 20 centimetres.
In triangle ACD: AC² + CD² = AD² So 225 plus CD² equals 289. This gives CD² equals 64, so CD equals 8 centimetres.
Finally, BD equals BC plus CD, which is 20 plus 8, giving 28 centimetres.
Before we conclude, let us recap the key ideas from today's lesson.
First, in a right-angled triangle, the hypotenuse is the side opposite the 90 degree angle, and it is always the longest side.
Second, the Pythagoras Theorem states that the square of the hypotenuse equals the sum of the squares of the other two sides.
Third, the converse of the theorem allows us to verify whether a triangle is right-angled by checking if the square of the longest side equals the sum of squares of the other two sides.
Fourth, by comparing the square of the longest side with the sum of squares of the other two sides, we can classify any triangle as right-angled, acute-angled, or obtuse-angled.
Fifth, always identify the hypotenuse correctly before applying the theorem — it is opposite the right angle.
Sixth, the theorem has beautiful applications in real-world problems involving distances, heights, and diagonal measurements.
That brings us to the end of our exploration of the Pythagoras Theorem. You now possess a powerful tool that has been used for over two thousand years to unlock the secrets of right-angled triangles. Keep practising, stay curious, and remember — mathematics is not about memorising formulas, but about understanding the beautiful relationships hidden in shapes and numbers. Until next time, happy learning!