Q. 14.2( 403 Votes )

Δ ABC and &

Answer :

It is given in the question that:

are two isosceles triangles

(i) In ,


AD = AD (Common)


AB = AC (Triangle ABC is isosceles)


BD = CD (Triangle DBC is isosceles)

 Three sides equal (SSS)Definition: Triangles are congruent if all three sides in one triangle are congruent to the corresponding sides in the other.

Therefore,

By SSS axiom,



(ii) In


AP = AP (Common)


PAB = PAC (By c.p.c.t)


AB = AC (Triangle ABC is isosceles)

Side-Angle-Side (SASRule. Side-Angle-Side is a rule used to prove whether a given set of triangles are congruent. If two sides and the included angle of one triangle are equal to two sides and included angle of another triangle, then the triangles are congruent

Therefore,

By SAS axiom,


(iii) PAB = PAC (By c.p.c.t)


AP bisects A (i)

Also,

In

PD = PD (Common)

BD = CD (Triangle DBC is isosceles)


BP = CP ( so by c.p.c.t)


Side-Side-Side (SSSRule. Side-Side-Side is a rule used to prove whether a given set of triangles are congruent. The SSS rule states that: If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent.

Therefore,

By SSS axiom,




BDP = CDP (By c.p.c.t) (ii)


By (i) and (ii), we can say that AP bisects A as well as D


(iv) BPD = CPD (By c.p.c.t)


And,


BP = CP (i)


Also,


BPD + CPD = 180o (BC is a straight line)


2BPD = 180o


BPD = 90o (ii)


From (i) and (ii), we get


AP is the perpendicular bisector of BC

Hence, Proved.

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
caricature
view all courses
RELATED QUESTIONS :

Prove that the anRS Aggarwal & V Aggarwal - Mathematics

If the sides of aRD Sharma - Mathematics

D is any point onNCERT Mathematics Exemplar

In a triangle RD Sharma - Mathematics

In Δ ABCRD Sharma - Mathematics

In Δ PQRRD Sharma - Mathematics

<span lang="EN-USRS Aggarwal & V Aggarwal - Mathematics

Prove that the peRD Sharma - Mathematics

In Fig. 10.25, <iRD Sharma - Mathematics

In a <span lang="RD Sharma - Mathematics