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PYTHAGORAS THEOREM
Pythagoras theorem is one of the first theorems known by people. It states that “in a right-angled triangle as {shown in figure 1}, the square of the hypotenuse is equal to the squares of the perpendicular and base”. In a right-angled triangle hypotenuse is the longest side. It is also called the Pythagorean theorem. The formula of Pythagoras theorem is given by; h^2 = p^2 + b^2 which is GK^2 =BK^2 +GB^2 Now let’s prove it. Let’s consider the above triangle We know, ∆GDB ~ ∆GBK Therefore,
G D
G B
G B
G K
[they are corresponding sides of a similar triangle]
Or, GB
2
=GD X GK………………………….[1]
K
B G
perpendicular[p]
base[b]
Hypotenuse[h]
K
h^ D
B G
b
p
Similarly, ∆BDK ~ ∆GBK Therefore,
K D
B K
=
B K
G K
[they are corresponding sides of a similar triangle] OR, BK^2 =KD X GK………………………….. [2] By adding equations [1] and [2] we get, GB^2 +BK^2 = GD X GK + KD X GK GB^2 +BK^2 =GK[GD+KD] Since, GD +KD =GK Therefore, GK^2 =GB^2 +BK^2 Hence, proved! we have proved the Pythagoras theorem Its applications and uses:
- It can be used to find the sides of the right-angled triangle.
- To confirm whether the triangle is a right-angled triangle or not.
- It can be used to find the diagonals of a square. Different methods to prove Pythagoras theorem:
- Proof using differentials
- Euclid’s proof
- Algebraic proof
- Proof using similar triangles References:
- Frohman, Charles. (2009). The Full Pythagorean Theorem.
- Pythagoras theorem, https://byjus.com/maths/pythagoras-theorem/
