Class 9 › Mathematics › Ganita Manjari I'm Up and Down, and Round and Round Class 9 Mathematics · Ganita Manjari · Chapter 5
Summary Questions & answers Practise this chapter → This chapter studies the circle and its many elegant properties. A circle is the set (locus) of all points in a plane at a fixed distance, the radius, from a centre. A chord joins two points on the circle; a chord through the centre is a diameter, the longest chord. Circles have rotational symmetry about the centre and reflection symmetry across every diameter. Infinitely many circles pass through two given points, with centres on the perpendicular bisector of the segment; exactly one circle (the circumcircle) passes through three non-collinear points. Key theorems: equal chords subtend equal angles at the centre and conversely; the line from the centre to a chord's midpoint is perpendicular to the chord and conversely; equal chords are equidistant from the centre and conversely; the longer of two chords lies nearer the centre. The angle an arc subtends at the centre is double the angle it subtends at any point on the remaining circle, so the angle in a semicircle is \(90^{\circ}\). Four points are concyclic when a segment subtends equal angles at two points on the same side, and opposite angles of a cyclic quadrilateral sum to \(180^{\circ}\).
Chord
A line segment joining any two points on a circle.
Diameter
A chord that passes through the centre; it is the longest chord, equal to twice the radius.
Arc
A connected portion of a circle between two points; the smaller is the minor arc, the larger the major arc.
Circumcircle
The unique circle passing through the three vertices of a triangle; its centre is the circumcentre.
Concyclic points
Points that all lie on a single circle.
Cyclic quadrilateral
A four-sided figure whose vertices all lie on one circle.
Q1 How is a circle defined? Answer Q2 How many circles pass through two given points? Answer Q3 How many circles pass through three non-collinear points? Answer Q4 What does the line from the centre to the midpoint of a chord do? Answer Q5 Equal chords of a circle are how placed relative to the centre? Answer Q6 Of two unequal chords, which is nearer the centre? Answer Q7 How is the angle at the centre related to the angle on the circle for the same arc? Answer Q8 What is the angle in a semicircle? Answer Q9 What is the sum of opposite angles of a cyclic quadrilateral? Answer Q10 Find the chord length if the radius is \(13\) cm and the chord is \(5\) cm from the centre. Answer Q1 Jamuna has a circular piece of paper. She is trying to locate its centre. Amina gives her a suggestion. She follows the instructions and is thrilled to find that it works. Can you guess what Amina told her? Answer Q1 1. What are the rotational symmetries of a square? How many lines of reflection symmetry does it have? What about a regular pentagon? A regular hexagon? Answer Q2 2. What is the length of the longest chord in a circle of radius 5 units? Is there a smallest chord? Answer Q3 3. The locus of points at a given distance from a given point is a circle. What can we say about the locus of points equidistant from two given points? (Hint: We know that any point that is equidistant from two given points A and B lies on the perpendicular bisector of AB. Does this make the perpendicular bisector the locus? For this, we have to show that all the points on the perpendicular bisector are equidistant from A and B.) Answer Q1 1. How many circles pass through two points on a plane? Answer Q2 2. Are there circles of all possible radii passing through A and B? What is the radius of the smallest circle passing through A and B? What is the radius of the largest circle passing through A and B? Answer Q3 3. As you move away from segment AB along its perpendicular bisector, do the radii of the circles containing A and B increase or decrease? Answer Q4 4. As you go along the perpendicular bisector, will the circle drawn from that point through A and B appear more curved or less curved? Answer Q5 5. You are given two points A and B on a plane. How many squares can you draw on the same plane with A and B on the boundary? How many squares can you draw on the plane with A and B as the corners of the square? Answer Q1 1. Draw ΔABC with AB = 5 cm, ∠A = 70° and ∠B = 60°. Draw the circumcircle of ΔABC. Is the centre inside or outside the triangle? Answer Q2 2. Draw ΔABC with AB = 5 cm, ∠A = 100°, AC = 4 cm. Draw the circumcircle of ΔABC. Is the centre inside or outside the triangle? Answer Q3 3. Draw ΔABC, with AB = 6 cm, BC = 7 cm and CA = 7 cm. Draw the circumcircle of ΔABC. Let the circumcentre be O. Measure OA, OB, OC. Answer Q4 4. What is the least possible radius of a circle through two points A and B? Answer Q1 1. A, B and C are three collinear points. Can you find a point P such that PA = PB = PC ? What can you say about the perpendicular bisectors of AB and BC? Draw and check. Can you show that for three collinear points A, B and C, the perpendicular bisector of AB and BC are parallel? Is it possible for a circle to pass through collinear points? Can you draw a line that cuts a given circle in three distinct points? Answer Q2 2. The circumcircle of a given ΔABC is drawn. Can there be other triangles congruent to ΔABC that share the same circumcircle? Answer Q1 1. Show that the triangle formed by a chord and the centre of the circle is isosceles. Answer Q2 2. Show that if two such isosceles triangles (occurring in the previous question) have equal base length, they are congruent to each other. Answer Q1 1. Can you explain why the converse to Theorem 4 is true, i.e., why does the perpendicular from the centre of a circle to a chord of the circle bisect the chord? (Hint: Use Fig. 5.12. You are told that ∠CMA = ∠CMB = 90°. You need to show that AM = BM.) Answer Q2 2. An isosceles triangle ABC is inscribed in a circle, with AB = AC. Show that the altitude from A to BC passes through the centre of the circle. Answer Q3 3. Two parallel chords of lengths 6 cm and 8 cm are on opposite sides of the centre of a circle. If the radius of the circle is 5 cm, find the distance between the midpoints of the chords. Answer Q1 1. Use the Baudhāyana–Pythagoras theorem to show why Theorem 6 must be true. Answer Q2 2. Consider Fig. 5.15. If CE is perpendicular to AB, CH is perpendicular to GH, and CE = CH, show that AB = GF. Answer Q3 3. Solve the previous question using the Baudhāyana–Pythagoras theorem. Answer Q1 1. Find the length of the chord of a circle where the radius is 7 cm and perpendicular distance is 6 cm. Answer Q2 2. Explain why the following statement is true: If the perpendicular distance of a chord from the centre is d and the radius is r, then the chord length is \(2 \sqrt{r^{2} - d^{2}}\). Answer Q3 3. In a circle, if the distance of chord AB from the centre is twice the distance of another chord CD from the centre, then can we conclude that CD = 2 AB? Give reasons for your answer. Answer Q1 A circle with centre O is drawn, and A, B, C, D are points on the circle (see Fig. 5.19). Measure the angles subtended by arc AKB and arc CLD at the centre O. If the angle at the centre is less than 180°, it is a minor arc. If the angle at the centre is greater than 180°, it is a major arc. State whether arcs AKB and CLD are minor arcs or major arcs. Answer Q1 1. In a circle with centre O, the central angle AOB is 60°. If the radius of the circle is 12 cm, what is the length of the chord AB? Answer Q2 2. Let A and B be two points on a circle with centre O. (i) Are there points X, Y on the circle, on the same side of AB, such that ∠AXB is different from ∠AYB? Answer Q3 2. Let A and B be two points on a circle with centre O. (ii) Is it true that if ∠AXB = ∠AYB, then X and Y lie on the same side of the circle? Answer Q4 2. Let A and B be two points on a circle with centre O. (iii) If ∠AXB = ∠AYB, and X and Y do not lie on the circle, does the circle through A, B and X also pass through Y? Answer Q5 3. Find x in Fig. 5.26. Answer Q1 A cyclic quadrilateral has angles measuring ∠A = 80°, ∠B = 110°, ∠C = 100°, and ∠D = 70°. Can such a quadrilateral be drawn? Explain why or why not. Answer Q1 1. In a circle, a chord is 5 cm away from the centre. If the radius of the circle is 13 cm, what is the length of the chord? Answer Q2 2. An arc of a circle subtends an angle of 70° at the centre. What is the measure of the angle subtended by the arc at a point on the circle? Answer Q3 3. The diameter of a circle is 26 cm. A chord of length 24 cm is drawn in the circle. Find the distance from the centre of the circle to the chord. Answer Q4 4. A circle has a radius of 15 cm. A chord is drawn. The distance from the centre of the circle to the chord is 9 cm. What is the length of the chord? Answer Q5 5. Prove that the perpendicular bisector of a chord passes through the centre of the circle. Answer Q6 6. The diameter of a circle is AB. Point C is on the circumference. What is the measure of the ∠ACB? Explain your reasoning. Answer Q7 7. ABCD is a cyclic quadrilateral inscribed in a circle. If ∠A measures 75°, what is the measure of ∠C? If ∠B measures 110°, what is the measure of ∠D? Answer Q8 8. Quadrilateral PQRS is inscribed in a circle. If ∠P = (2x + 10)° and ∠R = (3x − 20)°, find the value of x and the measures of ∠P and ∠R. Answer Q9 9. The distance of a chord of length 16 cm from the centre of a circle is 6 cm. Find the radius of the circle. Answer Q10 10. A cyclic quadrilateral has sides 5, 5, 12, 12 units. Find its area. Answer Q11 11. Consider a cyclic quadrilateral. Without drawing its circumcircle, how can we find out whether the centre of the circumcircle lies inside the quadrilateral or outside? What is the best way of finding out? Answer Q12 12. When two chords intersect, each of them is divided into two line segments. Show that if the intersecting chords are of equal length, then the line segments of one chord are equal to the corresponding line segments of the other chord. Answer Q13 13. Draw a circle in which a chord of 6 cm length stands at a distance of 3 cm from the centre. (Hint: Is it a circumcircle of a suitable triangle?) Answer Q14 14. Show that rectangle is the only parallelogram that can be inscribed in a circle. Answer Q15 15. Show that if a rectangle is inscribed in a circle, then the point of intersection of its diagonals must lie at the centre of the circle. Answer Q16 16. Consider all chords of a circle of a fixed length. What is the shape formed by the midpoints of all these chords? Answer Q17 17. In a circle with centre O, chords AB and AC are congruent. Explain why this statement is true: “The centre of the circle lies on the angle bisector of ∠BAC”. Answer Q18 18. Two parallel chords of lengths 10 cm and 24 cm are on the same side of the centre of a circle. The distance between the chords is 7 cm. Find the radius of the circle. Answer Q19 19. A regular hexagon is inscribed in a circle of radius r. Find the length of the sides of the hexagon and the distance of each side from the centre of the circle. Answer Q20 20. A quadrilateral MNOP is inscribed in a circle. If MN is a diameter, what can you say about ∠MOP and ∠MNP? Explain your reasoning. Answer Q21 21. Let ABCD be a cyclic quadrilateral. Explain why the exterior angle at any vertex is equal to the interior opposite angle (e.g., ∠CDE = ∠ABC, where E is a point on the extension of side CD). Answer Q22 22. “There is no chord of a circle that is longer than its diameter.” How do you justify this statement? Answer Q23 23. Let A be any point within a given circle with centre O. Show that the shortest chord of the circle that passes through point A is the one that is perpendicular to OA. Answer Q24 24. How would you use the following figure to justify the statement that the angle in a semicircle is 90°? Answer Q25 25. In a circle, two chords CC' and DD' are drawn perpendicular to a diameter AB. Prove that the segment MM' joining the midpoints of the chords CD and C' D' is perpendicular to AB. Answer Q26 26. How would you use the following figure to justify the statement that the sum of the opposite angles of a cyclic quadrilateral is 180°? Answer
A circle is the set of all points in a plane that are:
A equidistant from two lines B at a fixed distance from a centre C on a straight line D inside a square
Flashcards & interactive practice 
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