Question
triangle ABC is inscribed within a circle and has diameter line AB, with the inscribed angle at C. if the angle measure of A is 15 degrees, what is the arc measure of AC
Answers
GPT-4o mini
In a circle, when a triangle is inscribed, the angle subtended by a chord at the center of the circle is twice the angle subtended at any point on the circle. In your case, triangle ABC is inscribed in a circle with AB as the diameter. By the inscribed angle theorem, angle ACB is a right angle (90 degrees), since it subtends the semicircle defined by diameter AB.
Given that angle A measures 15 degrees, we can find the measure of angle B using the fact that the sum of angles in triangle ABC is 180 degrees:
\[
\angle A + \angle B + \angle C = 180^\circ
\]
\[
15^\circ + \angle B + 90^\circ = 180^\circ
\]
\[
\angle B = 180^\circ - 105^\circ = 75^\circ
\]
Now, we can find the arc measure of AC. The arc AC subtends angle B at the center of the circle. The angle subtended at the center by an arc is twice the angle subtended at any point on the circle; thus:
\[
\text{Arc AC} = 2 \times \angle B = 2 \times 75^\circ = 150^\circ
\]
Therefore, the arc measure of AC is **150 degrees**.
Given that angle A measures 15 degrees, we can find the measure of angle B using the fact that the sum of angles in triangle ABC is 180 degrees:
\[
\angle A + \angle B + \angle C = 180^\circ
\]
\[
15^\circ + \angle B + 90^\circ = 180^\circ
\]
\[
\angle B = 180^\circ - 105^\circ = 75^\circ
\]
Now, we can find the arc measure of AC. The arc AC subtends angle B at the center of the circle. The angle subtended at the center by an arc is twice the angle subtended at any point on the circle; thus:
\[
\text{Arc AC} = 2 \times \angle B = 2 \times 75^\circ = 150^\circ
\]
Therefore, the arc measure of AC is **150 degrees**.