Question

AD is the diameter of circumcircle of Δ ABC; AEBC.PA =AD​

Answer 1

Explanation:

Let a be the side of the equilateral triangle ABC.

∴ AB=BC=AC=a

and since ΔABC is an equilateral triangle

∠A=∠B=∠C=60

o

Given, radius of the circumcircle, r=6cm

Since the triangle ABC is equilateral, its perpendicular bisector i.e. the median meet at the same point O which is the centre of the in circle

∴ AD is the perpendicular bisector of BC.

⇒ BD=DC=

2

1

BC

=

2

a

and OB is the angle bisector of ∠B.

∴∠ABO =∠ OBD =

2

60

o

=30

o

In rt Δ ODB

sin30

o

=

OB

OD

⇒

2

1

=

r

OD

⇒OD=

2

r

⇒OD=

2

6

⇒OD=3cm

∴ Radius of the incircle =3cm.