Question

The hypotenuse is 10 and the area is 24, how do you find the other lengths on the triangle?

Answer 1

`6` and `8`.

Explanation:

Let the other side lengths be `a` and `b` and area be `A`. Then,
`color(white)xxA=24`
`=>(ab)/2=24color(white)(xxxxxxxxxxxxxxxx)`(area formula)
`=>2ab=96color(white)(xxxxxxxxxxxxxxxx)`

`=>a^2+b^2=100color(white)(xxxxxxxxxxxxx)`(The Pythagorean Theorem)

`=>a^2+b^2color(red)(+2ab)=100color(red)(+2ab)color(white)(xxxxxxxxxxxxx)`
`=>(a+b)^2=100+96color(white)(xxxxxxxxx)`(A fundamental identity)
`=>a+b=sqrt196color(white)(xxxxxxxxx)`
`=>a+b=14color(white)(xxxxxxxxx)`

The two number, of which product is `48` and sum is `14`, are `6` and `8`.