Question 1: Recall the formula for calculating the length of a chord when the radius and the perpendicular distance from the center to the chord are known.

Answer: Chord length = 2 * √(radius^2 - distance^2)

Question 2: If the radius of a circle is 10 cm and the perpendicular distance from the center to the chord is 6 cm, what is the length of the chord?

Answer: 2 * √(10^2 - 6^2) = 2 * √(100 - 36) = 2 * √64 = 2 * 8 = 16 cm

Question 3: Construct a circle of radius 8 cm and draw a chord that is 12 cm long. Use the grid to approximate and verify your construction.

Answer: Student's drawing and approximation; calculation involves verifying the chord length with the formula.

Question 4: Explain why the perpendicular bisector of a chord passes through the center of the circle.

Answer: Because the perpendicular bisector is the set of points equidistant from the endpoints of the chord, and all points on it are equally distant from the circle's center, which lies on this line.

Question 1: Calculate the length of a chord in a circle with radius 15 cm, if the perpendicular distance from the center to the chord is 9 cm.

Answer: 2 * √(15^2 - 9^2) = 2 * √(225 - 81) = 2 * √144 = 2 * 12 = 24 cm

Question 2: A chord of length 20 cm in a circle is 8 cm from the center. Find the radius of the circle.

Answer: radius = √( (chord length / 2)^2 + distance from center^2 ) = √(10^2 + 8^2) = √(100 + 64) = √164 ≈ 12.81 cm

Question 3: Construct a circle with radius 9 cm. Mark points A and B on the circle so that AB is a diameter. Draw a chord AB and find its length.

Answer: Diameter = 2 * radius = 18 cm

Question 4: A chord is 7 cm from the center of a circle with radius 10 cm. What is the length of the chord?

Answer: 2 * √(10^2 - 7^2) = 2 * √(100 - 49) = 2 * √51 ≈ 2 * 7.14 ≈ 14.28 cm

Question 5: In a circle, a chord is 5 cm away from the center and has length 12 cm. Find the radius of the circle.

Answer: radius = √( (chord length / 2)^2 + distance from center)^2 ) = √(6^2 + 5^2) = √(36 + 25) = √61 ≈ 7.81 cm

Question 1: A circle has radius 14 cm. A chord is 10 cm from the center. Find the length of the chord and explain why the perpendicular bisector of this chord passes through the center.

Answer: Chord length = 2 * √(14^2 - 10^2) = 2 * √(196 - 100) = 2 * √96 ≈ 2 * 9.80 ≈ 19.6 cm. The perpendicular bisector passes through the center because it is equidistant from both endpoints of the chord and aligns with the radius.

Question 2: A circle with radius 20 cm has a chord that is 16 cm from the center. Find the length of the chord and describe the steps involved.

Answer: Chord length = 2 * √(20^2 - 16^2) = 2 * √(400 - 256) = 2 * √144 = 2 * 12 = 24 cm.

Question 1: A circular fountain has a diameter of 18 meters. A walkway runs parallel to the diameter, 4 meters away from the center. What is the length of the walkway segment that crosses the fountain?

Answer: Radius = 9 m; distance from center to walkway = 4 m; chord length = 2 * √(9^2 - 4^2) = 2 * √(81 - 16) = 2 * √65 ≈ 2 * 8.06 ≈ 16.12 meters.

Question 2: A circular track has a radius of 50 meters. A spectator views a segment of the track where the chord length is 80 meters. Find the perpendicular distance from the center of the track to this chord.

Answer: distance = √(radius^2 - (chord length / 2)^2) = √(50^2 - 40^2) = √(2500 - 1600) = √900 = 30 meters

Question 1: Prove that the perpendicular bisector of any chord in a circle passes through the center, using a geometric argument.

Answer: By considering two points on the circle connected by a chord, the perpendicular bisector is equidistant from both points. Since all points on this bisector are equally distant from the endpoints, and the endpoints lie on the circle, this bisector must pass through the circle's center, which is equidistant from all points on the circle.

Question 2: In a circle of radius 12 cm, two chords are 9 cm and 6 cm away from the center. Find the lengths of both chords and compare their positions relative to the center.

Answer: Chord 1: length = 2 * √(12^2 - 9^2) = 2 * √(144 - 81) = 2 * √63 ≈ 2 * 7.94 ≈ 15.88 cm. Chord 2: length = 2 * √(12^2 - 6^2) = 2 * √(144 - 36) = 2 * √108 ≈ 2 * 10.39 ≈ 20.78 cm. The chord closer to the center (9 cm away) is shorter, as expected.

Question 1: A student states that the length of a chord is always greater than the radius of the circle. Is this correct? Explain and correct the statement if necessary.

Answer: Incorrect. The length of a chord can be less than, equal to, or greater than the radius, depending on the position of the chord. For example, a diameter equals twice the radius, but a chord very close to the center might be shorter than the radius.

Question 2: A calculation shows that the length of a chord in a circle with radius 10 cm is computed as 20 cm, but using the formula it should be 16 cm. Identify the mistake and provide the correct calculation.

Answer: Error: The calculation used the wrong value for the perpendicular distance or misapplied the formula. Correct calculation: for a perpendicular distance of 6 cm, chord length = 2 * √(10^2 - 6^2) = 16 cm.