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, what is the arc measure of AC
Answers
GPT-4o mini
By the properties of inscribed angles in a circle, the measure of an inscribed angle is half the measure of the arc it subtends. In triangle ABC, angle A subtends arc BC, and angle C subtends arc AB.
Given that angle A measures 15 degrees, we can find the measure of arc BC:
\[
\text{Arc } BC = 2 \times \text{Angle A} = 2 \times 15^\circ = 30^\circ
\]
The measure of the entire circle is 360 degrees. Since AB is the diameter of the circle, it divides the circle into two semicircles. Therefore, arc AB measures 180 degrees.
To find the measure of arc AC, we can use the fact that the total measure of the arcs around point A must equal 360 degrees:
\[
\text{Arc } AC + \text{Arc } BC = 180^\circ
\]
We already know that arc BC is 30 degrees, so we have:
\[
\text{Arc } AC + 30^\circ = 180^\circ
\]
Solving for arc AC:
\[
\text{Arc } AC = 180^\circ - 30^\circ = 150^\circ
\]
Thus, the measure of arc AC is \( \boxed{150} \) degrees.
Given that angle A measures 15 degrees, we can find the measure of arc BC:
\[
\text{Arc } BC = 2 \times \text{Angle A} = 2 \times 15^\circ = 30^\circ
\]
The measure of the entire circle is 360 degrees. Since AB is the diameter of the circle, it divides the circle into two semicircles. Therefore, arc AB measures 180 degrees.
To find the measure of arc AC, we can use the fact that the total measure of the arcs around point A must equal 360 degrees:
\[
\text{Arc } AC + \text{Arc } BC = 180^\circ
\]
We already know that arc BC is 30 degrees, so we have:
\[
\text{Arc } AC + 30^\circ = 180^\circ
\]
Solving for arc AC:
\[
\text{Arc } AC = 180^\circ - 30^\circ = 150^\circ
\]
Thus, the measure of arc AC is \( \boxed{150} \) degrees.