Question

In the given figure, L and M are the mid- points of AB and BC respectively.

(i) If AB = BC, prove that AL = MC.
(ii) If BL = BM, prove that AB = BC.

Hint
(i) $AB=BC\Rightarrow \frac{1}{2}AB=\frac{1}{2}BC\Rightarrow AL=MC$.
(ii) $BL=BM\Rightarrow 2BL=2BM\Rightarrow AB=BC$.

Answer 1

(i) It is given that L is the mid-point of AB.

∴ AL = BL = $\frac{1}{2}$AB .....(1)

Also, M is the mid-point of BC.

∴ BM = MC = $\frac{1}{2}$BC .....(2)

AB = BC (Given)

⇒ $\frac{1}{2}$AB = $\frac{1}{2}$BC (Things which are halves of the same thing are equal to one another)

⇒ AL = MC [From (1) and (2)]


(ii) It is given that L is the mid-point of AB.

∴ AL = BL = $\frac{1}{2}$AB

⇒ 2AL = 2BL = AB .....(3)

Also, M is the mid-point of BC.

∴ BM = MC = $\frac{1}{2}$BC

⇒ 2BM = 2MC = BC .....(4)

BL = BM (Given)

⇒ 2BL = 2BM (Things which are double of the same thing are equal to one another)

⇒ AB = BC [From (3) and (4)]