Question

In the above figure, O is the centre of the circle, FD is a tangent at C.Then, which of the following statements are true?

• A. $\angle ABC={90}^{\circ }$
• B. $\angle ACB={30}^{\circ }$
• C. $\angle ACD={90}^{\circ }$
• D. $\angle ABC={110}^{\circ }$

Answer 1

Answer: (A), (B), (C)

The correct options are
A

$\angle ABC={90}^{\circ }$

B

$\angle ACB={30}^{\circ }$

C

$\angle ACD={90}^{\circ }$

$\angle ACD={90}^{\circ }$ [Tangent at any point of a circle and the radius through this point are perpendicular to each other)
$\angle ACD=\angle ABC={90}^{\circ }$ [Angles in alternate segments are equal]
The angle between a tangent and a chord through the point of contact is equal to an angle in the alternate segment.

Consider $\Delta ABC$,
$\angle BAC+\angle ACB+\angle ABC={180}^{\circ }$
${60}^{\circ }+\angle ACB+{90}^{\circ }={180}^{\circ }$ [Given, $\angle BAC={60}^{\circ }$]
$\angle ACB=180-{150}^{\circ }\angle ACB={30}^{\circ }$
$\therefore$ Options (a), (b), (c) are true.