#### Question

The hypotenuse of a right triangle is `3sqrt10`. If the smaller leg is tripled and the longer leg doubled, new hypotenuse wll be `9sqrt5`. How long are the legs of the triangle?

#### Solution

Let the length of smaller side of right triangle be x cm then larger side be y cm

Then, as we know that by Pythagoras theorem

`x^2 + y^2 = (3sqrt10)^2`

x^{2} + y^{2} = 90 .............. (1)

If the smaller side is triple and the larger side be doubled, the new hypotenuse is `9sqrt5` cm

Therefore,

`(3x)^2+(2y)^2=(9sqrt5)^2`

9x^{2} + 4y^{2} = 405 ............. (2)

From equation (1) we get y^{2} = 90 - x^{2}

Now putting the value of y^{2} in equation (2)

9x^{2} + 4(90 - x^{2}) = 405

9x^{2} + 360 - 4x^{2} - 405 = 0

5x^{2} - 45 = 0

5(x^{2} - 9) = 0

x^{2} - 9 = 0

x^{2} = 9

`x = sqrt9` = ± 3

But, the side of right triangle can never be negative

Therefore, when x = 3 then

y^{2} = 90 - x^{2}

= 90 - (3)^{2}

= 90 - 9

= 81

`y=sqrt81`

= ± 9

Hence, length of smaller side of right triangle be 3 cm then larger side be 9 cm