Altitudes Ratio

Question

If h, h', h'' denote the lengths of the three altitudes of a triangle, which of the following ratios never occurs as the ratio h: h': h''?

a)2 : 3 : 4
b)2 : 3 : 5
c)2 : 4 : 5
d)3 : 4 : 5
e)3 : 4 : 6

Any help would be much appreciated! If possible, please could you explain the solution.

Thanks in Advance

Answer 1

Suppose that there is a triangle with altitudes $h_1$, $h_2$ and $h_3$.

Let $a_1$, $a_2$, and $a_3$ be the three sides of this triangle, such that the altitude from side $a_i$ is $h_i$. The area $A$ of the triangle is, by the well known formula: $$ A=\frac{1}{2}a_1h_1=\frac{1}{2}a_2h_2=\frac{1}{2}a_3h_3. $$ With this, we use the condition that for a triangle to exist, the sum of the length of its shorter sides must be bigger than the length of its longer side. In terms of altitudes, if $h_1\leq h_2\leq h_3$, the condition thus becomes $$ \frac{1}{h_1} \leq \frac{1}{h_2}+\frac{1}{h_3}, $$ which does not hold for b) only.