Question

ABC is a triangle and M & N are points on AB and AC such that MN || BC. Find the given ratios if AM:MB = 2:3.
1. $AN:AC$
2. $AN:NC$
3. $NC:AC$

Answer 1

According to BPT, if a line is drawn parallel to one side of a triangle to intersect the other two sides in different points then the other two sides are divided in the same ratio.

$\Rightarrow \frac{AM}{MB}=\frac{AN}{NC}$


$\text{Given}AM:MB=2:3⟹AN:NC=2:3$

$\frac{AN}{NC}=\frac{2}{3}$
Adding one on both sides, we get,

$\frac{AN}{NC}+1=\frac{2}{3}+1$

$\frac{AN+NC}{NC}=\frac{2+3}{3}$

$⟹\frac{AC}{NC}=\frac{5}{3}$

$⟹NC:AC=3:5$

$\frac{NC}{AN}=\frac{3}{2}$
Adding one on both sides, we get,

$\frac{NC}{AN}+1=\frac{3}{2}+1$

$\frac{NC+AN}{AN}=\frac{3+2}{2}$

$⟹\frac{AC}{AN}=\frac{5}{2}$

$⟹AN:AC=2:5$