For a right angled triangle
Side opposite to right angle is Hypotenuse
Side adjacent to right angle are Base & height
Note :
Here, we can also take AB as base & BC as height.
It does not change our answer
Let AB = a
BC = b
AC = c
Pythagoras theorem says that
Square of hypotenuse = Sum of square of other two sides
a ^{ 2 } + b ^{ 2 } = c ^{ 2 }
There are a lot of interesting things that we can do with Pythagoras theorem
Like
 Pythagoras triplets

Proof of Pythagoras theorem – The
normal way
Let’s do some questions
Find x
Here,
βABC is a right angled triangle, right angled at B
And,
AB = 3, BC = 4, AC = x
By Pythagoras Theorem
AC ^{ 2 } = AB ^{ 2 } + BC ^{ 2 }
x ^{ 2 } = 3 ^{ 2 } + 4 ^{ 2 }
x ^{ 2 } = 9 + 16
x ^{ 2 } = 25
x ^{ 2 } = 5 ^{ 2 }
Cancelling squares
x = 5
Therefore, x = 5
Find x
Here,
βABC is a right angled triangle, right angled at B
And,
AB = 6 cm, BC = 8 cm, AC = x
By Pythagoras Theorem
AC ^{ 2 } = AB ^{ 2 } + BC ^{ 2 }
x ^{ 2 } = 6 ^{ 2 } + 8 ^{ 2 }
x ^{ 2 } = 36 + 64
x ^{ 2 } = 100
x ^{ 2 } = 10 ^{ 2 }
Cancelling squares
x = 10 cm
Therefore, x = 10 cm
Find x
Here,
βPQR is a right angled triangle, right angled at P
And,
PR = 8 cm, PQ = 15 cm, QR = x
By Pythagoras Theorem
QR ^{ 2 } = PR ^{ 2 } + PQ ^{ 2 }
^{ }
Find x
Here,
βPQR is a right angled triangle, right angled at P
And,
PQ = 24 cm, QR = x, PR = 7 cm
By Pythagoras Theorem
QR ^{ 2 } = PQ ^{ 2 } + PR ^{ 2 }
x ^{ 2 } = (24) ^{ 2 } + 7 ^{ 2 }
x ^{ 2 } = 576 + 49
x ^{ 2 } = 625
x ^{ 2 } = (25) ^{ 2 }
Cancelling squares
x = 25 cm
Therefore, x = 25 cm
625 = 5 × 5 × 5 × 5
= 25 × 25
= (25) ^{ 2 }
Find x
In βABC,
AB = AC
Hence, it is an isosceles triangle.
Now, AD ⊥ BC
In isosceles triangle,
altitude and median are same
∴ AD is median of BC
i.e. D is midpoint of BC
∴ BD = DC = BC/2
BD = DC = x/2
In βADC, right angled at D.
By Pythagoras Theorem
(AB) ^{ 2 } = (AD) ^{ 2 } + (BD) ^{ 2 }
(37) ^{ 2 } + (12) ^{ 2 } + (x/2) ^{ 2 }
1369 = 144 + x ^{ 2 } /4
1369 − 144 = x ^{ 2 } /4
1225 = x ^{ 2 } /4
1225 × 4 = x ^{ 2 }
4900 = x ^{ 2 }
x ^{ 2 } = 4900
x ^{ 2 } = 70 ^{ 2 }
Cancelling squares
x = 70 cm
∴ x = 70 cm
Let's look at one last example