Question 1: Define an arc in a circle and state the key measurement you need to find when calculating an arc.

Answer: An arc is a part of the circumference of a circle; the key measurement is the angle or the measure of the arc in degrees or length.

Question 2: Calculate the measure of an arc that subtends a central angle of 60° in a circle with radius 10 cm.

Answer: 20π cm (if length of arc), or 60° (if measure in degrees)

Question 3: An arc has a length of 15 cm in a circle with radius 9 cm. What is the measure of the central angle in degrees?

Answer: Approximately 90°

Question 4: Identify the common mistake made when students confuse the measure of an arc with the measurement of the central angle.

Answer: Students often assume the arc length in degrees is equal to the central angle in degrees, which is incorrect unless the arc is a quarter of the circle.

Question 5: If a student calculates the arc length as 25 cm in a circle of radius 8 cm, what error might they have made? How should they correct it?

Answer: They likely used the formula length = r × θ (in radians) but did not convert θ correctly; they should ensure θ is in radians or use the degree version: arc length = (θ/360) × 2πr.

Question 6: Construct an arc in your diagram that subtends a 120° central angle in a circle of radius 7 cm. Label the arc measure.

Answer: The measure of the arc is 120°.

Question 7: A student claims that the length of an arc is always larger than the corresponding central angle in degrees. Is this true? Explain.

Answer: No, the arc length depends on the radius and the angle; for small angles and small radii, the arc length can be less than the angle in degrees.

Question 8: Given a circle with radius 12 cm and a central angle of 150°, calculate the arc length.

Answer: Approximately 31.42 cm (using arc length formula: (150/360)×2π×12)

Question 9: A diagram shows an arc with a length of 20 cm in a circle of radius 10 cm. What is the measure of the central angle in degrees? Show your working.

Answer: θ = (arc length)/(radius) = 20/10 = 2 radians ≈ 114.59°

Question 10: Identify and correct the misconception: 'The larger the arc, the smaller the central angle.'

Answer: Incorrect; the larger the arc length (or measure), the larger the central angle in degrees.

Question 11: A student calculates an arc length of 25 cm in a circle with radius 5 cm. The student then claims the central angle is 25°. Is this correct? Why or why not?

Answer: No; arc length in degrees is not equal to the measure of the central angle. They should convert arc length to angle: θ = (arc length / (2πr)) × 360.

Question 12: Challenge: Derive the formula for the arc length in terms of the radius and the central angle in degrees.

Answer: Arc length = (θ/360) × 2πr, where θ is the central angle in degrees.