1
A group of students is participating in a science fair. They have designed a model of a Ferris wheel. The wheel has a diameter of 20 meters.(a) If a point on the circumference of the Ferris wheel is at a height of 2 meters from the ground when it is at its lowest point, what is the height of the center of the Ferris wheel from the ground? [2 marks](b) What is the maximum height a person can reach on this Ferris wheel? [1 mark](c) A photographer wants to capture a picture of the Ferris wheel. He stands at a point P on the ground. If the angle subtended by the diameter of the Ferris wheel at point P is 90°, where must point P be located relative to the Ferris wheel? 2marks
2
Assertion: A circle is the locus of points equidistant from a fixed point.Reason: This fixed point is called the centre of the circle, and the constant distance is called the radius.(a) Assertion and Reason are true and Reason is the correct explanation of Assertion.(b) Assertion and Reason are true but Reason is not the correct explanation of Assertion.(c) Assertion is true, Reason is false.(d) Assertion is false, Reason is true. 1mark
3
A survey was conducted in a park to observe circular patterns. A researcher noted down the following observations:
- The path around a circular pond.
- The shape of a fountain's water spray at its peak.
- The boundary of a circular flower bed.
- The shape of a satellite dish.
The researcher wants to explain the property of symmetry in circles.
(a) How many lines of reflection symmetry does a circle have? [1 mark]
(b) Explain the rotational symmetry of a circle. [2 marks]
(c) If a circle has a radius of 7 cm, what is the length of its longest chord? [1 mark]
4
Consider two points A and B on a plane, 10 cm apart. A group of students are trying to draw circles that pass through both A and B.(a) Where do the centres of all such circles lie? [1 mark](b) If one of the circles drawn has a radius of 6 cm, what is the distance of its centre from the midpoint of AB? [2 marks](c) What is the radius of the smallest circle that can pass through A and B? 1mark
5
Three distinct points P, Q, and R are given on a plane.(a) If P, Q, and R are collinear, how many circles can pass through all three points? Explain why. [2 marks](b) If P, Q, and R are non-collinear, what can be said about the number of circles that can pass through them? [1 mark](c) The centre of the circle passing through three non-collinear points is called the ______ of the triangle formed by these points. 1mark
6
A craftsman is designing a circular decorative piece. He has two identical circular metallic rings. He wants to join them using a straight metallic rod.(a) If the two rings are of the same radius and placed such that they form a larger circular shape, and the rod connects two points on the circumference of the combined shape, what geometrical term can be used for this rod? [1 mark](b) If the rod is a chord of length 10 cm in one of the rings, and the angle subtended by this chord at the centre of the ring is 90°, find the radius of the ring. 3marks
7
In a school playground, there is a circular merry-go-round. A group of children are sitting on it.(a) If two children on the merry-go-round are at points A and B, and the angle subtended by the arc AB at the centre of the merry-go-round is 120°, what is the angle subtended by the same arc at a point C on the circumference (in the major arc)? [2 marks](b) If the merry-go-round has a diameter of 15 meters, and a child is sitting at point D on the circumference. If the angle subtended by the diameter at point D is 90°, does this statement provide any additional information about the position of D? Explain. 2marks
8
Rohan is sketching a design that involves a quadrilateral inscribed in a circle. He draws a quadrilateral ABCD such that its vertices lie on a circle.(a) If ∠A = 80°, what is the measure of ∠C? Justify your answer. [2 marks](b) If ∠B = 100°, what is the measure of ∠D? Justify your answer. [2 marks](c) Rohan then draws another quadrilateral PQRS, where ∠P = 95° and ∠R = 85°. Can this quadrilateral be inscribed in a circle? Explain. 2marks
9
A civil engineer is planning a circular water tank. The tank has a radius of 5 meters.(a) A pipe is to be laid along a chord of the tank. If the distance of the chord from the centre is 3 meters, what is the length of the chord? [3 marks](b) If another pipe is laid such that its length is 8 meters, what will be its distance from the centre of the tank? 2marks
10
A farmer is designing a circular field. He wants to divide the field using a straight fence.(a) If the fence is a chord of length 12 meters and the radius of the circular field is 10 meters, what is the distance of the fence from the centre of the field? [3 marks](b) If the farmer adds another fence of length 16 meters on the opposite side of the centre, what is its distance from the centre? 2marks
11
Assertion: Chords of a circle that are equidistant from the centre have equal length.Reason: In congruent isosceles triangles formed by a chord, centre, and radii, equal heights (distances from the centre) imply equal bases (chords).(a) Assertion and Reason are true and Reason is the correct explanation of Assertion.(b) Assertion and Reason are true but Reason is not the correct explanation of Assertion.(c) Assertion is true, Reason is false.(d) Assertion is false, Reason is true. 1mark
12
A group of students are studying the properties of arcs. They draw a circle and mark points A, B, and C on its circumference.(a) If arc AB subtends an angle of 100° at the centre, what is the angle subtended by arc AB at point C on the circumference (if C is on the major arc)? [2 marks](b) If arc BC subtends an angle of 80° at the centre, what is the angle subtended by arc BC at point A on the circumference (if A is on the major arc)? 2marks
13
A local artist is creating a circular mosaic. They use small circular tiles to form patterns.(a) If two chords in the mosaic are of equal length, how does their distance from the centre of the mosaic compare? [1 mark](b) If one chord is longer than another, which chord is closer to the centre of the mosaic? 1mark
14
Consider a quadrilateral ABCD inscribed in a circle.(a) If ∠A = 90°, what can be said about the side BD? [1 mark](b) If the quadrilateral is a rectangle, what is the relationship between its diagonals and the circle? 2marks
15
Rina is observing the moon on a particular night. The visible part of the moon appears circular.(a) Explain the symmetry of the moon's shape as observed. [2 marks](b) If Rina notices a spot on the edge of the moon that subtends an angle of 45° at the centre of the moon (as seen from her perspective), what angle does this spot subtend at any other point on the visible edge of the moon (on the opposite side of the centre)? 2marks
16
A surveyor is mapping a plot of land that has a circular boundary. There are three distinct trees at points P, Q, and R on this boundary.(a) Can the surveyor determine the exact location of the centre of the circular boundary using the positions of these three trees? Justify your answer. [2 marks](b) If the trees are located such that PQ = QR, and P, Q, R are on the circle, what can be said about the perpendicular bisector of PQ and QR? 1mark
17
Two parallel chords of lengths 6 cm and 8 cm are drawn in a circle of radius 5 cm. They are on opposite sides of the centre.(a) Calculate the distance of the 6 cm chord from the centre. [2 marks](b) Calculate the distance of the 8 cm chord from the centre. [2 marks](c) What is the total distance between the two chords? 1mark
18
Assertion: A quadrilateral with opposite angles summing to 180° is always cyclic.Reason: If a line segment subtends equal angles at two points on the same side of the line segment, then the four points are concyclic.(a) Assertion and Reason are true and Reason is the correct explanation of Assertion.(b) Assertion and Reason are true but Reason is not the correct explanation of Assertion.(c) Assertion is true, Reason is false.(d) Assertion is false, Reason is true. 1mark
19
A group of students are studying geometrical shapes. They are given a circle and asked to inscribe a triangle in it.(a) If the triangle inscribed in the circle is a right-angled triangle, where does the centre of the circle lie in relation to the triangle? [1 mark](b) If the triangle inscribed is an obtuse-angled triangle, where does the centre of the circle lie? 1mark
20
A designer is creating a logo that features a circle with two chords.(a) If the two chords subtend equal angles at the centre of the circle, what can be concluded about the lengths of the chords? [1 mark](b) If one chord is twice as long as another chord, how does its distance from the centre compare to the other chord's distance from the centre? (Assume the longer chord is not a diameter). 2marks
Answers
1
(a) The diameter is 20 m, so the radius is 10 m. When the lowest point is 2 m from the ground, the height of the centre is radius + lowest point height = 10 m + 2 m = 12 m. 2marks(b) Maximum height = diameter + lowest point height = 20 m + 2 m = 22 m. [1 mark](c) If the angle subtended by the diameter at point P is 90°, then by the converse of the angle in a semicircle theorem, point P must lie on the circumference of the circle. Therefore, P must be located on the circumference of the Ferris wheel. [2 marks]
2
(a) Assertion and Reason are true and Reason is the correct explanation of Assertion. 1mark
3
(a) A circle has infinitely many lines of reflection symmetry; every diameter is a line of symmetry. 1mark(b) A circle has complete rotational symmetry. It looks the same after rotation by any angle about its centre. [2 marks](c) The longest chord in a circle is its diameter. If the radius is 7 cm, the diameter is 2 * 7 cm = 14 cm. [1 mark]
4
(a) The centres of all circles passing through two given points A and B lie on the perpendicular bisector of the line segment AB. 1mark(b) Let M be the midpoint of AB. AM = MB = 5 cm. Let O be the centre of the circle with radius 6 cm. In right-angled triangle OMA, OA² = OM² + AM². So, 6² = OM² + 5². 36 = OM² + 25. OM² = 36 - 25 = 11. OM = √11 cm. [2 marks](c) The smallest circle passing through A and B has AB as its diameter. The radius would be half the length of AB, which is 10 cm / 2 = 5 cm. [1 mark]
5
(a) If P, Q, and R are collinear, no circle can pass through all three points. This is because if a circle passes through three points, they must form a triangle (be non-collinear). The perpendicular bisectors of PQ and QR would be parallel and would not intersect at a single point to form the centre of a circle. 2marks(b) If P, Q, and R are non-collinear, there is a unique circle that can pass through them. [1 mark](c) The centre of the circle passing through three non-collinear points is called the circumcentre of the triangle formed by these points. [1 mark]
6
(a) The rod can be referred to as a chord of the circular shape. 1mark(b) Let the radius of the ring be r. The chord length is 10 cm, and the angle subtended at the centre is 90°. Consider the triangle formed by the chord and the two radii to its endpoints. This is an isosceles triangle with two sides equal to r and the angle between them being 90°. By the Pythagorean theorem applied to the chord (which is the hypotenuse of the right-angled isosceles triangle formed by radii and chord, if the angle is 90 degrees at the centre), or by using the law of cosines: Chord² = r² + r² - 2rr*cos(90°). 10² = 2r² - 2r²(0). 100 = 2r². r² = 50. r = √50 = 5√2 cm. [3 marks]
7
(a) The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle. So, the angle subtended at point C is half the angle subtended at the centre. Angle at C = 100° / 2 = 50°. 2marks(b) The angle subtended by a diameter at any point on the circumference is always 90°. So, the statement that the angle subtended by the diameter at point D is 90° does not provide any additional information about the position of D; it simply confirms that D is on the circumference. [2 marks]
8
(a) In a cyclic quadrilateral, opposite angles are supplementary (add up to 180°). Therefore, ∠C = 180° - ∠A = 180° - 80° = 100°. 2marks(b) Similarly, ∠D = 180° - ∠B = 180° - 100° = 80°. [2 marks](c) Yes, quadrilateral PQRS can be inscribed in a circle because the sum of opposite angles ∠P + ∠R = 95° + 85° = 180°. Since one pair of opposite angles is supplementary, the quadrilateral is cyclic. [2 marks]
9
(a) Let the radius be r = 5 cm and the distance from the centre to the chord be d = 3 cm. Let the length of the chord be l. The relationship is given by (l/2)² + d² = r². (l/2)² + 3² = 5². (l/2)² + 9 = 25. (l/2)² = 16. l/2 = 4 cm. So, l = 8 cm. 3marks(b) If the pipe length (chord) is 8 meters, then from the previous calculation (l/2)² + d² = r². (8/2)² + d² = 5². 4² + d² = 25. 16 + d² = 25. d² = 9. d = 3 meters. [2 marks]
10
(a) Let the radius be r = 10 meters and the chord length be l = 12 meters. Let the distance from the centre be d. (l/2)² + d² = r². (12/2)² + d² = 10². 6² + d² = 100. 36 + d² = 100. d² = 64. d = 8 meters. 3marks(b) If the fence length (chord) is 16 meters, then (16/2)² + d² = 10². 8² + d² = 100. 64 + d² = 100. d² = 36. d = 6 meters. [2 marks]
11
(a) Assertion and Reason are true and Reason is the correct explanation of Assertion. 1mark
12
(a) The angle subtended by an arc at any point on the circumference (in the remaining part of the circle) is half the angle subtended at the centre. So, the angle at C = 100° / 2 = 50°. 2marks(b) Similarly, the angle subtended by arc BC at point A = 80° / 2 = 40°. [2 marks]
13
(a) If two chords in a circle are of equal length, they are equidistant from the centre of the circle. 1mark(b) The longer chord is closer to the centre of the circle. [1 mark]
14
(a) If ∠A = 90°, then since ABCD is a cyclic quadrilateral, the opposite angle ∠C = 180° - 90° = 90°. The sum of opposite angles ∠B and ∠D would also be 180°. If one angle of a cyclic quadrilateral is 90°, the opposite angle is also 90°. A quadrilateral with two opposite 90° angles implies that the diagonals subtend 90° at the circumference. If ∠A is 90°, then the arc BCD is a semicircle, which means BD is a diameter. 1mark(b) If the quadrilateral is a rectangle inscribed in a circle, its diagonals are diameters of the circle. The point of intersection of the diagonals is the centre of the circle. [2 marks]
15
(a) The moon's visible shape exhibits reflection symmetry across any diameter, and complete rotational symmetry about its centre. 2marks(b) The angle subtended by the segment connecting the two points on the visible edge (which form the arc) at the centre is 45°. The angle subtended by this arc at any other point on the visible edge of the moon (on the opposite side of the centre) will be half of the central angle, i.e., 45° / 2 = 22.5°. [2 marks]
16
(a) Yes, the surveyor can determine the exact location of the centre. The centre of the circle is the point of intersection of the perpendicular bisectors of the line segments joining any two pairs of these points (e.g., perpendicular bisector of PQ and QR). Since P, Q, and R are non-collinear, their perpendicular bisectors will intersect at a unique point, which is the circumcentre. 2marks(b) If PQ = QR, then their perpendicular bisectors will be equidistant from Q. Since the centre lies on both perpendicular bisectors, and they intersect at a single point, the centre is uniquely determined. This means the centre is equidistant from P, Q, and R. [1 mark]
17
(a) For the 6 cm chord: Let r = 5 cm. Half-chord length = 6/2 = 3 cm. Distance d1 from the centre: d1² + 3² = 5². d1² + 9 = 25. d1² = 16. d1 = 4 cm. 2marks(b) For the 8 cm chord: Half-chord length = 8/2 = 4 cm. Distance d2 from the centre: d2² + 4² = 5². d2² + 16 = 25. d2² = 9. d2 = 3 cm. [2 marks](c) Since the chords are on opposite sides of the centre, the total distance between them is the sum of their distances from the centre: d1 + d2 = 4 cm + 3 cm = 7 cm. [1 mark]
18
(a) Assertion and Reason are true and Reason is the correct explanation of Assertion. (The reason explains how to prove the converse of Theorem 10, which is also related to cyclic quadrilaterals). 1mark
19
(a) If the triangle inscribed in the circle is a right-angled triangle, the centre of the circle lies at the midpoint of the hypotenuse. 1mark(b) If the triangle inscribed is an obtuse-angled triangle, the centre of the circle lies outside the triangle. [1 mark]
20
(a) If two chords subtend equal angles at the centre of the circle, then the lengths of the chords are equal. 1mark(b) Let the length of the shorter chord be l1 and its distance from the centre be d1. Let the length of the longer chord be l2 = 2*l1 and its distance from the centre be d2. We know that longer chords are closer to the centre. Thus, d2 < d1. While a direct ratio isn't given, the relationship between chord length and distance from the centre is inverse. The longer chord (l2) will be closer to the centre (d2 < d1). [2 marks]
