For a right angled triangle
Β
Side opposite to right angle is Hypotenuse
Side adjacent to right angle are Base & height
Β
Note :
Here, we can also take AB as base & BC as height.
It does not change our answer
LetΒ Β AB = a
BC = b
AC = c
Pythagoras theorem says that
Β Square of hypotenuse = Sum of square of other two sides
Β Β a 2 + b 2 = c 2
Β
There are a lot of interesting things that we can do with Pythagoras theorem
Like
- Pythagoras triplets
-
Proof of Pythagoras theorem β The
normal way
Letβs do some questions
Β
Find x
Here,
Β βABC is a right angled triangle, right angled at B
Β
And,
Β AB = 3, BC = 4, AC = x
Β
By Pythagoras Theorem
AC 2 = AB 2 + BC 2
Β
x 2 = 3 2 + 4 2
x 2 = 9 + 16
x 2 = 25
x 2 = 5 2
Cancelling squares
Β x = 5
Β
Therefore, x = 5
Β
Find x
Here,
Β βABC is a right angled triangle, right angled at B
Β
And,
Β AB = 6 cm, BC = 8 cm, AC = x
Β
By Pythagoras Theorem
AC 2 = AB 2 + BC 2
Β
x 2 = 6 2 + 8 2
x 2 = 36 + 64
x 2 = 100
x 2 = 10 2
Cancelling squares
Β x = 10 cm
Β
Therefore, x = 10 cm
Β
Find x
Here,
Β βPQR is a right angled triangle, right angled at P
Β
And,
Β PR = 8 cm, PQ = 15 cm, QR = x
Β
By Pythagoras Theorem
QR 2 = PR 2 + PQ 2
Β
Find x
Here,
Β βPQR is a right angled triangle, right angled at P
Β
And,
Β PQ = 24 cm, QR = x, PR = 7 cm
Β
By Pythagoras Theorem
QR 2 = PQ 2 + PR 2
Β Β Β Β Β x 2 = (24) 2 + 7 2
Β Β Β Β Β x 2 = 576 + 49
Β Β Β Β Β x 2 = 625
Β Β Β Β Β x 2 = (25) 2
Cancelling squares
Β x = 25 cm
Β
Therefore, x = 25 cm
625 = 5 Γ 5 Γ 5 Γ 5
Β Β Β Β = 25 Γ 25Β
Β Β Β Β = (25) 2
Β
Β
Β
Β
Find x
In βABC,
Β AB = AC
Hence, it is an isosceles triangle.
Β
Now, AD β₯ BC
Β
In isosceles triangle,
altitude and median are same
Β
β΄ AD is median of BC
i.e. D is mid-point of BC
β΄ BD = DC = BC/2
Β Β BD = DC = x/2
Β
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
Β
1369 β 144 = x 2 /4
1225 = x 2 /4
Β Β 1225 Γ 4 = x 2
Β Β Β Β Β 4900 = x 2
Β Β Β Β Β Β Β x 2 = 4900
Β
Β Β Β Β Β x 2 =Β 70 2
Cancelling squares
x = 70 cm
Β
β΄ x = 70 cm
Let's look at one last example
Β
Get live Maths 1-on-1 Classs - Class 6 to 12