Question

If A, B, C, D are four points such that $\angle BAC={30}^{\circ }and\angle BDC={60}^{\circ }$ then

prove that

D is the centre of the circle through A, B and C

Answer 1

R.E.F image

Now, we know the theorem

The angle which an are of a circle

Subtends at the center is double which

it subtends at any point on the

remaining part of the circumference.

Hence according to the above theorem

$\angle BOC=2\angle BAC$

Now,

$\angle BDC={60}^{\circ }$ [Given]

$=2\times {30}^{\circ }$

$\angle BDC=2\times \angle BAC$ [$\because \angle BAC={30}^{\circ }$ Given]

Hence, from the above theorem, we

can say that D is the center of

the circle through $A,B$ and $C$