In Fig. 15.82, ABC and ABD are two triangles on the base AB. If line segment CD is bisected by AB at O, Show that ar(Δ ABC) = ar(Δ ABD).

Question

In Fig. 15.82, ABC and ABD are two triangles on the base AB. If line segment CD is bisected by AB at O, Show that ar(&#x394; ABC) = ar(&#x394; ABD).

Philoid · Accepted Answer

Given that,

CD bisected AB at O

To prove: Area () = Area ()

Construction: CP perpendicular to AB and DQ perpendicular to AB

Proof: Area ( = (AB * CP) (i)

Area ( = (AB * DQ) (ii)

In , we have

∠CPO = ∠DQO (Each 90^o)

Given that,

CO = DO

∠COP = ∠DOQ (Vertically opposite angle)

Then, by AAS congruence rule

Therefore,

CP = DQ (By c.p.c.t)

Thus,

Area ( = Area ()

Hence, proved