Points A and B are marked on a circle centered at point O so that the angle AOB = 45. The length of the greater

Points A and B are marked on a circle centered at point O so that the angle AOB = 45. The length of the greater of the arcs into which the circle is divided by points A and B is 133. Find the length of the smaller arc.

1. Since ∠AOB = 45 ° central, the degree measure of the smaller arc AB is 45 ° (since the degree measure of the central angle is equal to the degree measure of the arc on which it rests).
Thus, the degree measure of the larger arc AB is equal to:
Arc AB> = 360 ° – 45 ° = 315 °.
2. The arc length is found by the formula:
l = (π * R * α) / 180 °,
where R is the radius of the circle, α is the degree measure of the arc.
By hypothesis l AB> = 133, and since the arc AB> = 315 °, then:
(π * R * 315 °) / 180 ° = 133;
π * R * 315 ° = 180 ° * 133;
R = (180 ° * 133) / (π * 315 °) = 76 / π.
3. Find the length of the smaller arc AB:
l AB <= (π * 76 / π * 45 °) / 180 ° = (76 * 45 °) / 180 ° = 76/4 = 19.
Answer: The length of the smaller arc AB is 19.