Question

circle relationships: mastery test
point o is the center of the circle in the diagram. what is ( mangle bca )?
a. ( 80^{circ} )
b. ( 75^{circ} )
c. ( 70^{circ} )
d. ( 65^{circ} )

Explanation:

Step1: Find the measure of ∠BOA

The sum of angles around a point is \(360^{\circ}\). Given one angle is \(250^{\circ}\), so \(\angle BOA=360^{\circ}- 250^{\circ}=110^{\circ}\).

Step2: Use the property of tangents and radii

Since \(OA\perp CA\) and \(OB\perp CB\) (radius is perpendicular to the tangent at the point of tangency), \(\angle OBC = \angle OAC=90^{\circ}\).
In quadrilateral \(OBCA\), the sum of interior angles is \((4 - 2)\times180^{\circ}=360^{\circ}\). Let \(\angle BCA=x\). Then \(90^{\circ}+90^{\circ}+110^{\circ}+x = 360^{\circ}\).
Simplify the equation: \(290^{\circ}+x=360^{\circ}\), so \(x = 360^{\circ}-290^{\circ}=70^{\circ}\).

Answer:

C. \(70^{\circ}\)