Question

A right triangle has one leg that is 5 cm longer than the other leg and the hypotenuse is 25 cm long. How do you find the length of each leg?

Answer 1

No such triangle exists.

Explanation:

If we denote the shorter leg as `x` then
`color(white)("XXX")x^2+(x+5)^2=25^2`

`rarr color(white)("XXXX")2x^2+10x=0`

`rarr color(white)("XXXX")x(x+10)=0`

`rarr color(white)("XXXX")x=0 or x=-10`

Since neither of these are possible for any real triangle,
no triangle exists that meets the given conditions.

Answer 2

15cm and 20cm.

Explanation:

Since it is a right triangle use`color(blue) " Pythagoras's Theorem"`

which states 'in a right triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides'

This can be written as an equation. If c is the hypotenuse and a and b are the other 2 sides ( the legs ) then

`c^2 = a^2 + b^2`

Here , let one of the legs be x , so the other is x + 5

then `25^2 = x^2 + (x+5)^2 " using theorem "`

now distribute brackets and collect like terms. Also going to reverse the equation so x terms are on the left side.

hence : `x^2 + x^2 + 10x +25 = 625`

so `2x^2 + 10x +25 = 625`

this is a quadratic equation , hence equate to zero to solve.
`2x^2 + 10x - 600 = 0`
common factor of 2 :`2(x^2 + 5x -300) = 0`

Require factors of -300 which sum to 5 (coefficient of x term). These are +20 and - 15 . If unsure use quadratic formula to obtain them.

`rArr 2(x+20)(x-15) = 0`

the solution is x = - 20 or x = 15 : x > 0 thus x = 15

The 2 legs are x and x+5 hence 15 and 20.