The diameter of circle C is 18 cm. is 60°. Use this information, the diagram, and your experiences in geometry this semester to answer these questions. (Use .)
a) Find . Explain.
b) Find . Explain.
c) Find . Explain.
d) Find the length L of . Show your work.
What is 60º? We don't have the diagram to help you.
a) To find angle , we need to use the fact that the angle subtended by an arc at the center of a circle is twice the angle subtended by the same arc at any point on the circumference of the circle.
Since the diameter of circle C is 18 cm, the radius is half of the diameter, which is 9 cm. Let's label the center of the circle as point O, and any point on the circumference as point A.
Since angle is subtended by arc AC, we know that angle is twice angle . Therefore, angle = 2 * 60° = 120°.
b) To find angle , we need to use the fact that angles on a straight line add up to 180 degrees.
Angle and angle are both subtended by arc AC, and since they are on the same straight line, their sum is equal to 180°.
Therefore, angle = 180° - angle = 180° - 60° = 120°.
c) To find angle , we need to use the fact that angles in the same segment of a circle are equal.
Angle is subtended by arc AC, and angle is subtended by the same arc. Since they are in the same segment of the circle, they are equal.
Therefore, angle = 60°.
d) To find the length L of arc AC, we need to use the formula for calculating the length of an arc:
L = ( / 360°) * 2 * π * r
where is the measure of the angle in degrees, and r is the radius of the circle.
In this case, = 60° and r = 9 cm (since the diameter is 18 cm and the radius is half of the diameter).
L = (60° / 360°) * 2 * π * 9 cm
L = (1/6) * 2 * π * 9 cm
L = (1/3) * π * 9 cm
L = π * 3 cm
L ≈ 9.42 cm
Therefore, the length of arc AC is approximately 9.42 cm.