Find the ratio of the sides of a triangle if:

(a) the angles are in the ratio 1:2:3

(b) the angles are in the ratio 4:5:6

Hi sinjid9,

if the ratio of angles is 1:2:3 there are 6 parts and the sum of the angles in a triangle is 180 degrees.

One angle is \(\displaystyle \frac{180^o}{6}=30^o\)

The other angles are twice and three times that, \(\displaystyle 60^o,\ 90^o\)

To calculate the ratios of the sides, you can re-arrange the Sine Rule (Law of Sines).

\(\displaystyle \frac{SinA}{a}=\frac{SinB}{b}=\frac{SinC}{c}\)

to get

\(\displaystyle \frac{SinA}{SinB}=\frac{a}{b}\)

then a:b=SinA:SinB

Also \(\displaystyle \frac{SinB}{b}=\frac{SinC}{c}\)

\(\displaystyle \frac{SinB}{SinC}=\frac{b}{c}\)

then a:b:c=SinA:SinB:SinC

When the ratio of the angles is 4:5:6, there are 15 parts

\(\displaystyle \frac{180^o}{15}=12^o\)

Then the angles are \(\displaystyle 4(12^o),\ 5(12^o),\ 6(12^o)\)

then solve as above