IN THE GIVEN FIGURE, 'O' IS THE CENTRE OF THE CIRCLE IF BD=DC AND ANGLE DBC=30 FIND THE MEASURES - Maths - Circles

Question

IN THE GIVEN FIGURE, 'O' IS THE CENTRE OF THE CIRCLE IF BD=DC AND
ANGLE DBC=30 FIND THE MEASURES OF ANGLE BAC AND ANGLE BOC?

Varun.Rawat · Accepted Answer

In ∆BDC, we have    BD = DC  Given⇒∠BCD = ∠DBC  Angles opposite to equal sides in a ∆ are equal⇒∠DBC = 30°In ∆BDC,∠BCD + ∠DBC + ∠BDC = 180° Angle sum property⇒30° + 30° + ∠BDC = 180° ⇒ ∠BDC = 120°Since, ABDC is a cyclic quadrilateral, so∠BDC + ∠BAC = 180°   Opposite angles of cyclic quad
 are supplementary⇒120° + ∠BAC = 180°⇒∠BAC = 60°We know that angle subtended by an arc at the centre is double the angle subtended by the same arc at any point on the circle
So, ∠BOC = 2∠BAC = 2×60 = 120°

IN THE GIVEN FIGURE, 'O' IS THE CENTRE OF THE CIRCLE.IF BD=DC AND ANGLE DBC=30. FIND THE MEASURES OF ANGLE BAC AND ANGLE BOC?