Question

Given a right triangle with sides of length a, b, and c and area a²+b²-c², find c/b, the ratio of the legs of the right triangle.

Answer 1

4

Explanation:

Looking at the sketch we can formulate 2 equations:

Pythag:
`b^(2) +c^(2) =a^(2)`

Area of triangle=
`(1)/(2) bc`

But area also =
`a^(2) +b^(2) -c^(2)` (from the question)

Therefore
`(1)/(2) bc` =
`a^(2) +b^(2) -c^(2)`

But we can sub in the expression for
`a^(2)` from pythag. (
`b^(2) +c^(2) =a^(2)`)

`(1)/(2) bc= b^(2) +c^(2) +b^(2) -c^(2)`

`(1)/(2) bc=2b^(2)`

`bc=4b^(2)`

`c=4b`

`c/b= 4b/b=4`