Question

In the figure below, AB and AC are two tangents of the circle from the same external point A touching at points B and C, respectively. If ∠BAC = 2z° and ∠BDC = 5z°, then find the value of ∠BOC + ∠BAC in degrees.

• A. 105
• B. 135
• C. 140
• D. 155
• E. 160

Answer 1

The Correct Option is B

Join a line from B to C.

\(∠ABC = ∠BDC = 5z°\) [Alternate segment theorem]
Similarly, \(∠BCA = ∠CDB = 5z°\)
Thus, in \(△\;ABC\),
\(∠ABC + ∠BCA + ∠BAC = 180°\)
\(z = 15°\)
\(5z + 5z + 2z = 180°\)
So, \(∠BDC = 5(15) = 75°\)
Thus, \(∠BOC = 180° – 75° = 105°\) (As DBOC is a cyclic quadrilateral)
\(∠BAC = 2(15) = 30°\)
Hence, \(∠BOC + ∠BAC = 105° + 30° = 135°\)

Hence, option B is the correct answer.