Question

In triangle $$ABC$$ ; $$AB=AC.\ P,Q$$ and $$R$$ are mid-points of sides $$AB,AC$$ and $$BC$$ respectively. Prove that:
$$PR=QR$$

Answer 1

In $$\triangle ABC,$$
$$AB=AC$$ [Given]
$$\Rightarrow \dfrac{1}{2}AB=\dfrac{1}{2}AC$$
$$\Rightarrow AP=AQ$$......(i) [Since $$P$$ and $$Q$$ are mid-points]
In $$\triangle BCA,$$
$$PR=\dfrac{1}{2}AC$$ [$$PR$$ is line joining the mid-points of $$AB$$ and $$BC$$]
$$\Rightarrow PR=AQ$$......(ii)
In $$\triangle CAB,$$
$$QR=\dfrac{1}{2}AB$$ [$$QR$$ is line joining the mid-points of $$AC$$ and $$BC$$]
$$\Rightarrow QR=AP$$......(iii)
From (i)(ii) and (iii)
$$PR=QR$$