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) = ar(Δ MBC) (iii) ar(Δ ABM) = ar(Δ ACL) (iv) ar(Δ LOB) = ar(Δ MOC)

Question

In a &#x394; ABC, if L and M are points on AB and AC respectively such that LM||BC. Prove that: (i) ar(&#x394; LCM) = ar(&#x394; LBM) (ii) ar(&#x394; LBC) = ar(&#x394; MBC) (iii) ar(&#x394; ABM) = ar(&#x394; ACL) (iv) ar(&#x394; LOB) = ar(&#x394; MOC)

Philoid · Accepted Answer

(i) Clearly, triangles LMB and LMC are on the same base LM and between the same parallels LM and BC.

Therefore,

Area ( = Area () (1)

(ii) We observe that triangles LBC and MBC are on the same base BC and between the same parallels LM and BC.

Therefore,

Area (ΔLBC) = Area (ΔMBC) (2)

(iii) We have,

Area ( = Area ( [From (i)]

Area ( + Area ( = Area ( + Area (

Area ( = Area (

(iv) We have,

Area ( = Area ( [From (ii)]

Area (LBC) - Area ( = Area ( - Area (

Area ( = Area ()

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) = ar(Δ MBC)(iii) ar(Δ ABM) = ar(Δ ACL)(iv) ar(Δ LOB) = ar(Δ MOC)

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) = ar(Δ MBC)

(iii) ar(Δ ABM) = ar(Δ ACL)

(iv) ar(Δ LOB) = ar(Δ MOC)