The sides of a right angled triangle are in AP. Show that . . .

Question : The sides of a right angled triangle are in AP. Show that they are in the ratio 3:4:5.

Doubt by Kanika

Solution : 

Let the three consecutive terms of AP which are also the sides of the right angled triangle be (a-d), (a) and (a+d)

Since hypotenuse is the longest side of a right angled triangle. Hence, (a+d) must be the hypotenuse and the (a-d) & (a) must be other two sides. 

Using Pythagoras Theorem

H²=B²+P²

(a+d)²=(a-d)²+(a)²
a²+d²+2ad=a²+d²-2ad+a²
2ad=-2ad+a²

2ad+2ad=a²
4ad=a²
4ad/a=a
4d=a

Hence, a=4d

Hence, the three terms of the AP or sides will be 
a₁=a-d
a₁=4d-d
a₁=3d

a₂=a
a₂=4d

a₃=a+d
a₃=a+4d
a₃=5d

Now the ratio of a₁, a₂ and a₃ would be 
a₁:a₂:a₃=3d:4d:5d
a₁:a₂:a₃=3:4:5

Hence, the three sides of a right angled triangle are in the ratio of 3:4:5.