# If the angles of the triangle are in the ratio 3:4:5, then how do you find the ratio of the sides?

Sep 23, 2015

#### Answer:

(Assuming I haven't messed up)
The ratio of the sides (to 4 significant digits) is

$1 : 1.225 : 1.366$

#### Explanation:

Part 1
For any triangle with sides $a , b , c$ and corresponding opposite angles $A , B , C$ the Law of Sines tells us that

$\frac{\sin \left(A\right)}{a} = \frac{\sin \left(B\right)}{b} = \frac{\sin \left(C\right)}{c}$

or, by rearranging:

$\frac{b}{a} = \frac{\sin \left(B\right)}{\sin \left(A\right)}$

and

céa=sin(C)/sin(A)

Part 2
If $\angle A : \angle B : \angle C$ are in the ratio $3 : 4 : 5$
since $\angle A + \angle B + \angle C = \pi$

$\angle A = \frac{3 \pi}{12} = \frac{\pi}{4}$

$\angle B = \frac{4 \pi}{12} = \frac{\pi}{3}$

$\angle C = \frac{5 \pi}{12}$

Part 3
$a : b : c$
$\textcolor{w h i t e}{\text{XXX}} = \frac{a}{a} : \frac{b}{a} : \frac{c}{a}$

$\textcolor{w h i t e}{\text{XXX}} = 1 : \sin \frac{B}{\sin} \left(A\right) : \sin \frac{C}{\sin} \left(A\right)$

$\textcolor{w h i t e}{\text{XXX}} = 1 : \sin \frac{\frac{\pi}{3}}{\sin} \left(\frac{\pi}{4}\right) : \sin \frac{\frac{5 \pi}{12}}{\sin} \left(\frac{\pi}{4}\right)$

(and using my calculator):
$\textcolor{w h i t e}{\text{XXX}} = 1 : 1.224745 : 1.366025$