Question

In a triangle $ABC,$ altitudes from vertices $A$ and $B$ have lengths $3$ and $6$ respectively. Then the exhaustive set of values of the length of altitude from vertex $C$ is

• A. $(3,7)$
• B. $(2,6)$
• C. $(3,4)$
• D. $(1,5)$

Answer 1

The correct option is B $(2,6)$


Let altitudes from vertices $A,B$ and $C$ have lengths $h_a,h_b$ and $h_c$ respectively.
Then $h_a=3$ and $h_b=6$
We know that
$|a-b|<c<a+b$
$\Rightarrow |\frac{2\Delta }{h_a}-\frac{2\Delta }{h_b}|<\frac{2\Delta }{h_c}<\frac{2\Delta }{h_a}+\frac{2\Delta }{h_b},$
where $\Delta$ denotes the area of triangle $ABC.$
$\Rightarrow |\frac{1}{3}-\frac{1}{6}|<\frac{1}{h_c}<\frac{1}{3}+\frac{1}{6}$
$\Rightarrow \frac{1}{6}<\frac{1}{h_c}<\frac{1}{2}$
$\Rightarrow h_c\in (2,6)$