Question 1: Calculate the length of an arc in a circle with radius 10 cm and a central angle of 60°. Use π ≈ 3.14.

Answer: 10 * (60/360) * 2 * 3.14 * 10 = 10.47 cm

Question 2: A sector of a circle has an arc length of 15 cm and a radius of 5 cm. Find the measure of the central angle in degrees.

Answer: (15 / (2 * 3.14 * 5)) * 360 ≈ 171.97°

Question 3: Construct an arc that subtends a 90° angle at the center of a circle with radius 8 cm. Calculate the arc length.

Answer: 8 * (90/360) * 2 * 3.14 * 8 = 12.56 cm

Question 4: In a circle, an arc measures 120° and the radius is 15 cm. Determine the length of the arc.

Answer: 15 * (120/360) * 2 * 3.14 * 15 = 39.4 cm

Question 5: A student makes a common mistake by using 180° instead of 360° when calculating the arc length. Identify and correct this error for an arc with a 45° angle and radius 9 cm.

Answer: Incorrect: (45/180) * 2 * 3.14 * 9 = 3.58 cm (error: should divide by 360, not 180). Correct: (45/360) * 2 * 3.14 * 9 = 7.15 cm

Question 6: Explain how the length of an arc relates to the measure of its central angle in a circle.

Answer: The length of an arc is directly proportional to the measure of its central angle; specifically, it is (angle/360) times the circle’s circumference.

Question 7: A sector of a circle has an area of 50 cm² and a radius of 10 cm. Find the measure of the central angle in degrees.

Answer: Sector area = (θ/360) * π * r² → 50 = (θ/360) * 3.14 * 100 → θ = (50 * 360) / (3.14 * 100) ≈ 57.32°

Question 8: Plot an arc with a radius of 12 cm that subtends a 75° angle at the center. Then, calculate the arc length.

Answer: 12 * (75/360) * 2 * 3.14 * 12 = 23.56 cm

Question 9: A circle has an arc of 90° and a radius of 20 cm. An object moves along this arc. How much distance does it cover?

Answer: 20 * (90/360) * 2 * 3.14 * 20 = 62.8 cm

Question 10: Identify the mistake in the following calculation: An arc of 45° in a circle with radius 10 cm is calculated as (45/180) * 2 * 3.14 * 10.

Answer: The mistake is dividing by 180 instead of 360. Correct approach: (45/360) * 2 * 3.14 * 10.

Question 11: Challenge: A sector with a radius of 9 cm has an arc length of 7.07 cm. Find the central angle in degrees and verify the calculation.

Answer: θ = (7.07 / (2 * 3.14 * 9)) * 360 ≈ 45° (since 2 * 3.14 * 9 ≈ 56.52, (7.07/56.52)*360 ≈ 45°)