In a circle centered at O, a diameter AB is extended to a point C outside the circle. CD is a tangent at point D on the circle. If CD = 7√3 cm and ∠DBC = 120°, then the length of the diameter of the circle is equal to:
Show answer & explanation
Correct answer: C
Explanation
The correct answer is option 3 i.e. 14 cm.
The radius of a circle is perpendicular to the tangent line at the point of tangent.
Let the radius = OD = OB = OA = r cm
∠ODC = 90°
∠DBO = 180° - 120° = 60°
Also, ∠ODB = 60° (In triangle ODB, OD = OB)
Now, ∠DOB = 60° (Triangle ODB is an equilateral triangle)
∠DCO = 180° - ∠ODC - ∠DOC = 180° - 90° - 60° = 30°
tan 30° = OD/CD
⇒ 1/√3 = r/7√3
⇒ r = 7
⇒ 2r = 14
Therefore, the length of the diameter of the circle is 14 cm.