Question

If $Q$ is mid-point of line segment $AB$ and $P$ is mid-point of $QB$, then show that $PB=\frac{1}{4}AB$

Answer 1

$Q$ is the midpoint of $\overline{AB}\Rightarrow \overline{AQ}+\overline{QB}=\overline{AB}$

$\Rightarrow \overline{AB}=2\overline{AQ}=2\overline{QB}$ .........$\left(1\right)$

$P$ is the midpoint of $\overline{QB}\Rightarrow \overline{QP}+\overline{PB}=\overline{QB}$

$\Rightarrow \overline{QB}=2\overline{QP}=2\overline{PB}$ .........$\left(2\right)$

From $\left(1\right)$ and $\left(2\right)$ we have

$\overline{AB}=2\overline{QB}$ from $\left(1\right)$

$\overline{AB}=2\left(2\overline{PB}\right)$ from $\left(2\right)$

$\Rightarrow \overline{AB}=4\overline{PB}$

$\therefore \overline{PB}=\frac{1}{4}\overline{AB}$