Question

In the given figure, AB = AC.

Prove that:
(i) DP = DQ
(ii) AP = AQ
(iii) AD bisects angle A

Answer 1

Const: Join AD.

In ΔABC,
AB = AC .......[Given]
∴ ∠C = ∠B  ......(i) [angles opp. to equal sides are equal]

(i)
In ΔBPD and ΔCQD,
∠BPD = ∠CQD ......[Each = 90°]
∠B = ∠C ........[proved]
BD = DC ........[Given]
∴ ΔBPD ≅ ΔCQD .......[AAS criterion]
∴ DP = DQ ......[c.p.c.t]

(ii) 
We have already proved that ΔBPD ≅ ΔCQD
Therefore, BP = CQ .......[c.p.c.t]
Now,
AB = AC .......[Given]
⇒ AB − BP = AC − CQ
⇒ AP = AQ

(iii)
In  ΔAPD and ΔAQD,
DP = DQ .......[proved]
AD = AD .......[common]
AP = AQ ........[Proved]
∴ ΔAPD ≅ ΔAQD ......[SSS]
⇒ ∠PAD = ∠QAD .......[c.p.c.t]

Hence, AD bisects angle A.