Question

Given that $BC=12,D$ is the mid point of $AB$ and $E$ divides $BC$ in the ratio $4:1$ ratio externally and $F$ divides $AC$ internally in the ratio $m:n$ ($m$ and $n$ are co-prime). Then the value of $m+n$ is

• A. $4$
• B. $5$
• C. $6$
• D. $7$

Answer 1

The correct option is B $5$

$D$ is the midpoint of $AB$
$\Rightarrow AD=DB$
$\frac{BE}{CE}=4\Rightarrow \frac{BE-CE}{CE}=3\Rightarrow BC=3CE\Rightarrow CE=4,BE=16$

Using Menelaus' theorem,
$\frac{AD}{DB}\times \frac{BE}{CE}\times \frac{CF}{FA}=1$
$\Rightarrow \frac{16}{4}\times \frac{n}{m}=1$
$\Rightarrow \frac{m}{n}=\frac{4}{1}$
$\Rightarrow m+n=4+1=5$

*Alternative solution
Let $B(0,0),C(12,0),\&A(6,6)$
$\Rightarrow E\equiv (16,0)\&D\equiv (3,3)$



By solving line $DE$ and $AC$
$\Rightarrow F\equiv (\frac{54}{4},\frac{6}{5})\Rightarrow \frac{m}{n}=\frac{4}{1}$