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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Eric Bizzell
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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`
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