The hypotenuse of a right triangle is 6m more than the twice of the shortest side. If the third side is 2m less than the hypotenuse, find the sides of the triangle.

Explanation

Let the shortest side of the triangle be x cm.
Then, its hypotenuse `=(2x+6)cm.`
And, its third side `=(2x+6-2)cm=(2x+4)cm.`
By Pythagoras' theorem, we have
`(2x+6)^(2)=x^(2)+(2x+4)^(2)`
`implies" "4x^(2)+24x+36=5x^(2)+16x+16`
`implies" "x^(2)-8x-20=0`
`implies" "x^(2)-10x+2x-20=0`
`implies" "x(x-10)+2(x-10)=0implies(x-10)(x+2)=0`
`impliesx-10=0" or "x+2=0impliesx=10" or "x=-2`
`implies" "x=10" "[because" the side of a triangle cannot be negative"]`
`:." ""shorter side "=10 cm," longer side "=(2xx10+4)cm=24cm`
and hypotenuse `=(2xx10+6)cm=26cm.`
Hence, the sides of the given triangle are 10 cm, 24 cm and 26 cm.