Question

In the given figure, AB is a diameter and AC is a chord of the circle and the tangent at C intersects AB produced in D. If $\angle BAC={30}^{\circ }$, then $\angle CDB=$ ______.

• A. ${30}^{\circ }$
• B. ${60}^{\circ }$
• C. ${100}^{\circ }$
• D. ${120}^{\circ }$

Answer 1

The correct option is A

${30}^{\circ }$

$\angle ACB={90}^{\circ }$ (angles in the semicircle)
$\angle BCD={30}^{\circ }$ ( angles in the alternate segment are equal)
$\angle ACD=\angle ACB+\angle BCD$
$={90}^{\circ }+{30}^{\circ }$$={120}^{\circ }$
$\angle CDB={180}^{\circ }-({30}^{\circ }+{120}^{\circ })$ .....(sum of angles in a triangle)
$={180}^{\circ }-{150}^{\circ }$$={30}^{\circ }$