Can someone help me with this excersize? The ratio of heights to the sides of the triangle is v_{a}:v_{b}:v_{c}=12:5:8. What are the length of the sides of triangle? (a,b,c=?) Thank you!
I tried using formula for area of triangle, e.g. P=(av_a)/2 and P=(bv_b)/2 and then I would get b=12/5 a and I did that for other combinations but did not come up with anything.
Yes, your idea works!
Since $$S=\frac{ah_a}{2}=\frac{bh_b}{2}=\frac{ch_c}{2},$$ we obtain: $$\frac{1}{a}:\frac{1}{b}:\frac{1}{c}=12:5:8$$ or $$a:b:c=\frac{1}{12}:\frac{1}{5}:\frac{1}{8}.$$ We need to check that $$\frac{1}{12}+\frac{1}{8}>\frac{1}{5},$$ of course.
For minimal natural's $a$, $b$ and $c$ we obtain $(10,24,15).$