Right Angled Triangle solve knowing the ratio and C

Question

Equation for a Right Angled Triangle

$C = \sqrt{A^2 + B^2}$

If you know $C$

$C = 49$

And you know the ratio of width to height

$16:9$

Is it possible to work out $A$ and $B$ or will there be too many answers. I can't imagine there will be because the ratio makes it pretty strict.

Answer 1

WLOG, let $A > B$

Then the ratio implies that $9A = 16B\implies A=\frac{16}{9}B$

Substituting this in:

$$49=\sqrt{(\frac{16}{9}B)^2+B^2} = B\sqrt{\frac{337}{81}} $$

$$B = \frac{49\cdot 9}{\sqrt{337}} = \frac{441}{\sqrt{337}}$$

Then $A = \frac{784}{\sqrt{337}}$

Answer 2

The legs are 16k and 9k for some positive constant k.

Then, $(16k)^2 + (9k)^2 = 49^2$

From which, the value of k can be found. Hence, the lengths of the legs can also be found.

Answer 3

As we can see,

$C^2=A^2+B^2$

It means C is largest side and equal to 49. And two sides A and B are in ratio 16:9.

Let A=16x, B=9x.

Put value of A, B and C to find values.