Scale sides of a triangle

Question

If I have a triangle with sides A B and C, how can I scale the triangle down to one that has sides A + B = 1?

E.g, if I have the triangle ABC where length of A = 45, length of B = 55 and length of C = sqrt(45^2 + 55^2), how can I scale that down to a triangle where A + B = 1? Especially if A + B doesnt conveniently add up to 100.

Sorry, i've not come across this problem before.

It strikes me as something quite simple, like similar triangles, i just cant see it

Answer 1

You are looking to scale down a triangle by dividing each side by some value $x$ such that

$$\frac Ax+\frac Bx=\frac{A+B}x=1$$.

It quickly follows that $x=A+B$.

Answer 2

Given your triangle $ABC$, you are looking for a triangle $A' B' C'$ such that $A' + B' = 1$ and you want to preserve similarity such that $\frac{A + B}{C} = \frac{A'+B'}{C'}$ which is equal to $\frac{1}{C'}$. So you've got three equations and three unknowns :)