# If the sides of a right angled triangle are in AP, then the sides of the acute angles are:The options a. sq.rt 3,1/sq.rt 3 b.sq.rt(sq.rt5-1)/2, sq.rt(sq.rt5+1)/2 c.3/5,4/5 d.none of these...

If the sides of a right angled triangle are in AP, then the sides of the acute angles are:

The options

a. sq.rt 3,1/sq.rt 3

b.sq.rt(sq.rt5-1)/2, sq.rt(sq.rt5+1)/2

c.3/5,4/5

d.none of these

Please provide the answer with solution...

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The sides of a right triangle are in arithmetic progression.Let one of the legs be x, the other leg x+d, and the hypotenuse x+2d.

Then by the Pythagorean theorem we have:

`x^2+(x+d)^2=(x+2d)^2`

`x^2+x^2+2dx+d^2=x^2+4dx+4d^2`

`3d^2+2dx-x^2=0`

`(3d-x)(d+x)=0`

`==>3d=x` or `d=-x`

We want the ratio of the legs or `x/(x+d)`

If `d=-x` this makes no sense in the problem as one of the sides has length 0.

`3d=x==>x/(x+d)=(3d)/(3d+d)=3/4`

So the ratio of sides is 3:4.

**The answer is (c) `3/5,4/5` **

**Note that this is a Pythagorean triple: `3/5,4/5,5/5` with common difference `1/5`