In figure, ABC and ABD are two triangles on the base AB. If line segme

1.	In figure, ABC and ABD are two triangles on the base AB. If line segment CD is bisected by AB at O, show that ar(ΔABC) = ar(ΔABD).
Answer» Draw two perpendiculars CP and DQ on AB. Now, ar(ΔABC) = 1/2 x AB x CP ⋅⋅⋅⋅⋅⋅⋅(1) ar(ΔABD) = 1/2 x AB x DQ ⋅⋅⋅⋅⋅⋅⋅(2) To prove the result, ar(ΔABC) = ar(ΔABD), we have to show that CP = DQ. In right angled triangles, ΔCPO and ΔDQO ∠CPO = ∠DQO = 90° CO = OD (Given) ∠COP = ∠DOQ (Vertically opposite angles) By AAS condition: ΔCP0 ≅ ΔDQO So, CP = DQ …………..(3) (By CPCT) From equations (1), (2) and (3), we have ar(ΔABC) = ar(ΔABD) Hence proved.

In figure, ABC and ABD are two triangles on the base AB. If line segment CD is bisected by AB at O, show that ar(ΔABC) = ar(ΔABD).

Answer»

Draw two perpendiculars CP and DQ on AB.

Now,

ar(ΔABC) = 1/2 x AB x CP ⋅⋅⋅⋅⋅⋅⋅(1)

ar(ΔABD) = 1/2 x AB x DQ ⋅⋅⋅⋅⋅⋅⋅(2)

To prove the result, ar(ΔABC) = ar(ΔABD), we have to show that CP = DQ.

In right angled triangles, ΔCPO and ΔDQO

∠CPO = ∠DQO = 90°

CO = OD (Given)

∠COP = ∠DOQ (Vertically opposite angles)

By AAS condition:

ΔCP0 ≅ ΔDQO

So, CP = DQ …………..(3) (By CPCT)

From equations (1), (2) and (3), we have

ar(ΔABC) = ar(ΔABD)

Hence proved.