Question
Angle BCD is a circumscribed angle of circle A. Angle BAC measures 53°.
Circle A is shown. Line segments B A and D A are radii. Tangents B C and D C intersect at point C outside of the circle. A line is drawn to connect points A and C. Angle B A C is 53 degrees.
What is the measure of angle BCD?
37°
53°
74°
106°
Circle A is shown. Line segments B A and D A are radii. Tangents B C and D C intersect at point C outside of the circle. A line is drawn to connect points A and C. Angle B A C is 53 degrees.
What is the measure of angle BCD?
37°
53°
74°
106°
Answers
GPT-4o mini
To find the measure of angle BCD, we can use the property of tangents and inscribed angles.
In this scenario, angle BAC is an inscribed angle that intercepts arc BC of circle A. The measure of an inscribed angle is half the measure of the arc it intercepts. Therefore, the arc BC measures twice the angle BAC:
\[
\text{Arc BC} = 2 \times \text{Angle BAC} = 2 \times 53^\circ = 106^\circ.
\]
Since angle BCD is a circumscribed angle that intercepts the same arc BC, the measure of angle BCD is half the measure of arc BC. Therefore:
\[
\text{Angle BCD} = \frac{1}{2} \times \text{Arc BC} = \frac{1}{2} \times 106^\circ = 53^\circ.
\]
This gives us 53°, which corresponds to the inscribed angle BAC.
However, the circumscribed angle BCD actually corresponds to the remaining part for the exterior angle:
\[
\text{Angle BCD} = 360^\circ - 53^\circ = 106^\circ.
\]
Thus, the correct measure of angle BCD is:
\[
\boxed{106^\circ}.
\]
In this scenario, angle BAC is an inscribed angle that intercepts arc BC of circle A. The measure of an inscribed angle is half the measure of the arc it intercepts. Therefore, the arc BC measures twice the angle BAC:
\[
\text{Arc BC} = 2 \times \text{Angle BAC} = 2 \times 53^\circ = 106^\circ.
\]
Since angle BCD is a circumscribed angle that intercepts the same arc BC, the measure of angle BCD is half the measure of arc BC. Therefore:
\[
\text{Angle BCD} = \frac{1}{2} \times \text{Arc BC} = \frac{1}{2} \times 106^\circ = 53^\circ.
\]
This gives us 53°, which corresponds to the inscribed angle BAC.
However, the circumscribed angle BCD actually corresponds to the remaining part for the exterior angle:
\[
\text{Angle BCD} = 360^\circ - 53^\circ = 106^\circ.
\]
Thus, the correct measure of angle BCD is:
\[
\boxed{106^\circ}.
\]