Circle Tangent Angle Problem

This solution details how to find the measure of angle OBC when tangents are drawn to a circle from an external point.

Given Tangent Information

• Distance from external point A to center O: AO = 6 cm.
• Radius of the circle: OB = OC = 3 cm.
• AB and AC are tangents to the circle from point A.

Key Geometric Properties

• The radius to the point of tangency is perpendicular to the tangent line ($\angle ABO = 90^\circ$).
• Tangents from an external point are equal in length (AB = AC).
• The line segment from the center to the external point (AO) bisects the angle between the tangents ($\angle BAC$) and the angle formed by radii to the points of tangency ($\angle BOC$).
• Triangle OBA is a right-angled triangle (at B).

Angle OBC Calculation

1. Analyze Right Triangle OBA:

   Consider the right-angled triangle OBA.

   • Hypotenuse AO = 6 cm.
   • Side OB = 3 cm.

   Use trigonometry to find $\angle OAB$. The side OB is opposite to $\angle OAB$.

   $ \sin(\angle OAB) = \frac{Opposite}{Hypotenuse} = \frac{OB}{AO} = \frac{3 \text{ cm}}{6 \text{ cm}} = \frac{1}{2} $

   From the sine value, we find:

   $ \angle OAB = 30^\circ $

2. Determine Angle ABC:

   Since AO bisects $\angle BAC$, the angle $\angle BAC = 2 \times \angle OAB$.

   $ \angle BAC = 2 \times 30^\circ = 60^\circ $

   In $\triangle ABC$, AB = AC (tangents from A), so it's an isosceles triangle.

   The base angles are equal: $\angle ABC = \angle ACB$.

   The sum of angles in $\triangle ABC$ is $180^\circ$.

   $ \angle BAC + \angle ABC + \angle ACB = 180^\circ $

   $ 60^\circ + 2 \times \angle ABC = 180^\circ $

   $ 2 \times \angle ABC = 180^\circ - 60^\circ = 120^\circ $

   $ \angle ABC = \frac{120^\circ}{2} = 60^\circ $

3. Calculate Angle OBC:

   We know that the radius OB is perpendicular to the tangent AB.

   $ \angle ABO = 90^\circ $

   Angle ABO can be split into two parts: $\angle OBC$ and $\angle ABC$.

   $ \angle ABO = \angle OBC + \angle ABC $

   Substitute the known values:

   $ 90^\circ = \angle OBC + 60^\circ $

   Solve for $\angle OBC$:

   $ \angle OBC = 90^\circ - 60^\circ = 30^\circ $

The measure of angle OBC is $30^\circ$.