Question

angle bcd is a circumscribed angle of circle a. angle bca measures 40°. what is the measure of minor arc bd?
○ 40°
○ 50°
○ 80°
○ 100°

Explanation:

Step1: Identify right triangles

Since $CB$ and $CD$ are tangents to circle $A$, $\angle ABC = \angle ADC = 90^\circ$.

Step2: Find $\angle BAC$

In $\triangle ABC$, sum of angles is $180^\circ$.
$\angle BAC = 180^\circ - 90^\circ - 40^\circ = 50^\circ$

Step3: Use tangent congruence property

Tangents from a point to a circle are congruent, so $\triangle ABC \cong \triangle ADC$. Thus $\angle DAC = \angle BAC = 50^\circ$.

Step4: Calculate $\angle BAD$

$\angle BAD = \angle BAC + \angle DAC = 50^\circ + 50^\circ = 100^\circ$

Step5: Relate central angle to arc

The measure of a minor arc equals its central angle. So minor arc $BD = \angle BAD$.

Answer:

100°