Question

In Fig. 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).

Answer 1

Given that, 

CD bisected AB at O

To prove : 

Area (ΔABC) = Area (ΔABD)

Construction : 

CP perpendicular to AB and DQ perpendicular to AB

Proof : 

Area (ΔABC) = \(\frac{1}{2}\)(AB x CP) ...(i)

Area (ΔABD) = \(\frac{1}{2}\)(AB x DQ) ...(ii)

In ΔCPO and ΔDQO, 

We have,

∠CPO = ∠DQO 

(Each 90°) 

Given that, 

CO = DO 

∠COP = ∠DOQ

(Vertically opposite angle) 

Then, 

by AAS congruence rule,

ΔCPO ≅ ΔDQO

Therefore, 

CP = DQ 

(By c.p.c.t) 

Thus,

Area (ΔABC) = Area (ΔABD)

Hence, proved