Question

In given figure, P is the mid-point of BC and Q is the mid-point of AP. If BQ when produced meets AC at R, Prove that $RA=\frac{1}{3}CA$.

Answer 1

Given A $△ABC$ in which P is the mid-point of BC, Q is the mid-point of BC, Q is the mid-point of AP, such that BQ produced meets AC at R

To prove $RA=\frac{1}{3}CA$

Construction Draw PS||BR, meeting AC at S.

Proof In $△BCR$, P is the mid-point of BC and PS||BR.

$\therefore$ $Sisthemid-pointofCR.$

$\Rightarrow$ $CS=SR$

In $△APS$, Q is the mid-point of AP and QR||PS.

$\therefore$ $Risthemid-pointofAS$

$\Rightarrow$ $AR=RS$

From (i) and (ii), we get

$AR=RS=SC$

$\Rightarrow$ $AC=AR+RS+SC=3AR$

$\Rightarrow$ $AR=\frac{1}{3}AC=\frac{1}{3}CA$ $\left[Henceproved\right]$