If the bisectors of the interior angle $A$ of $\triangle ABC$ divides $BC$ into segments $BD = 4, DC = 2$. If the length of the altitude $AD$ is $2\sqrt{10}$ and if $AB$ and $AC$ are integers. Then the possible length of the side $AC$ is/are:
Step-by-Step Solution
Key Concept: The ratio $c:b = 2:1$ directly constrains the relationship between all geometric parameters through coordinate equations.
Given $\frac{c}{b} = 2$, so $c = 2b$, we have $c^2 = 4b^2$. From $c^2 = b^2 + k^2$, we get $k^2 = 3b^2$. From $b^2 = (h-8)^2 + k^2$, substituting yields $b^2 = (h-8)^2 + 3b^2$, giving $(h-8)^2 = -2b^2$. Since $k^2 \geq 10$, we have $3b^2 \geq 10$, and combining with constraints gives $11 < b^2 < 30$.
Correct Answer: 2,4