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Quantitative Ability and Data Interpretation Question on Circles, Chords and Tangents

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.
AB and AC are two tangents of the circle from the same external point A

A

105

B

135

C

140

D

155

E

160

Answer

135

Explanation

Solution

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

Hence, option B is the correct answer.