Question

In a Δ ABC, if L and M are points on AB and AC respectively such that LM || BC. Prove that:

(i) ar (Δ LCM) = ar (Δ LBM)

(ii) ar (Δ LBC) = (Δ MBC)

(iii) ar (Δ ABM) = ar (Δ ACL)

(iv) ar (Δ LOB) = ar (Δ MOC)

Answer 1

Given:

In ΔABC, if L and M are points on AB and AC such that LM||BC

To prove:

(i)

(ii)

(iii)

(iv)

Proof: We know that triangles between the same base and between the same parallels are equal in area.

(i) Here we can see that ΔLMB and ΔLMC are on the same base BC and between the same parallels LM and BC

Therefore

…… (1)

(ii) Here we can see that ΔLBC and ΔLMC are on the same base BC and between the same parallels LM and BC

Therefore

…… (2)

(iii) From equation (1) we have,

(iv) From (2) we have,