Class 9 Mathematics • Ganita Manjari • Chapter 5 — I'm Up and Down, and Round and Round
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इस page पर कुल 30 Important Questions हैं।
A. Circle Basics and Symmetry
Important Question 1
Circle, Centre and Radius
What is a circle? In a circle with centre O, if OA = 7 cm and A lies on the circle, find the radius and the distance from O to any other point B on the circle.
आसान हिंदी में:\n\nCircle के हर boundary point की centre से दूरी बराबर होती है। OA boundary तक की दूरी है, इसलिए वही radius है।
हिंदी में Step-by-Step Solution:\n\n
Given OA = 7 cm. Since OA is a radius, radius = 7 cm. Every point on the circle is 7 cm from O, so OB = 7 cm.
✅ Radius = 7 cm and OB = 7 cm.
Important Question 2
Radius and Diameter
A circle has radius 9 cm. Find its diameter. If the diameter is 18 cm, is it also a chord?
आसान हिंदी में:\n\nDiameter हमेशा centre से होकर जाने वाला chord होता है और उसका length radius का double होता है।
हिंदी में Step-by-Step Solution:\n\n
Diameter = 2 × radius = 2 × 9 = 18 cm. A diameter joins two points of the circle and passes through the centre, so it is also a chord.
✅ Diameter = 18 cm; Yes, every diameter is a chord.
Important Question 3
Identify the Chord
A segment PQ joins two points P and Q on a circle but does not pass through the centre. Is PQ a chord, a diameter, or a radius? Explain.
आसान हिंदी में:\n\nदो boundary points को जोड़ने वाला segment chord होता है। Centre से गुजरना जरूरी नहीं है; centre से गुजरने पर ही वह diameter बनता है।
हिंदी में Step-by-Step Solution:\n\n
P and Q are both on the circle, so PQ is a chord. Because it does not pass through the centre, it is not a diameter. It is not a radius because it does not start at the centre.
✅ PQ is a chord, not a radius or diameter.
Important Question 4
Locus of a Point
A point moves so that its distance from a fixed point O is always 5 cm. What is the locus of the point?
आसान हिंदी में:\n\nकिसी fixed point से fixed distance पर घूमने वाले सभी points एक circle बनाते हैं।
हिंदी में Step-by-Step Solution:\n\n
All points whose distance from O is exactly 5 cm form a circle with centre O and radius 5 cm.
✅ The locus is a circle with centre O and radius 5 cm.
Important Question 5
Reflection Symmetry
How many lines of reflection symmetry does a circle have? What kind of lines are they?
आसान हिंदी में:\n\nCircle को उसके centre से गुजरने वाली किसी भी line के बारे में reflect करने पर circle वही रहता है।
हिंदी में Step-by-Step Solution:\n\n
Every diameter of a circle is a line of reflection symmetry. Since there are infinitely many diameters, a circle has infinitely many lines of reflection symmetry.
✅ Infinitely many; every diameter is a reflection-symmetry line.
B. Circles Through Points and Circumcircle
Important Question 6
Circles Through Two Points
Two fixed points A and B are 10 cm apart. Can more than one circle pass through A and B? Where will the centres of such circles lie?
आसान हिंदी में:\n\nदो points से गुजरने वाले circle के centre की दूरी A से और B से equal होगी। ऐसे centres perpendicular bisector पर lie करते हैं।
हिंदी में Step-by-Step Solution:\n\n
Yes. Infinitely many circles can pass through two fixed points A and B. Their centres lie on the perpendicular bisector of AB.
✅ Yes, infinitely many; centres lie on the perpendicular bisector of AB.
Important Question 7
Three Non-collinear Points
Three non-collinear points A, B and C are given. How many circles can pass through all three points?
आसान हिंदी में:\n\nतीन non-collinear points के लिए perpendicular bisectors of AB and AC एक unique point पर intersect करते हैं। वही centre बनता है।
हिंदी में Step-by-Step Solution:\n\n
The perpendicular bisectors of two sides of triangle ABC meet at one point, the circumcentre. Hence exactly one circle passes through A, B and C.
✅ Exactly one circle — the circumcircle of triangle ABC.
Important Question 8
Circumcentre
In triangle ABC, the perpendicular bisectors of AB and AC intersect at O. What is O called, and what is true about OA, OB and OC?
आसान हिंदी में:\n\nTriangle की sides के perpendicular bisectors का intersection circumcentre कहलाता है। वह तीनों vertices से equal distance पर होता है।
हिंदी में Step-by-Step Solution:\n\n
O is the circumcentre of triangle ABC. Therefore OA = OB = OC, and this common length is the circumradius.
✅ O is the circumcentre and OA = OB = OC.
Important Question 9
Unique Circle Condition
Could three collinear points determine one unique circle? Give a reason based on the chapter idea of a circumcircle.
आसान हिंदी में:\n\nUnique circumcircle की condition three non-collinear points है। तीन collinear points के perpendicular bisectors ऐसा single circumcentre निर्धारित नहीं करते।
हिंदी में Step-by-Step Solution:\n\n
No. The chapter states the unique-circle result for three non-collinear points. Three collinear points do not determine a finite-radius circumcircle through all three points.
✅ No; the three points must be non-collinear.
C. Chords and Their Properties
Important Question 10
Equal Chords and Central Angles
In the same circle, chord AB = chord CD. If angle AOB = 70°, where O is the centre, find angle COD.
आसान हिंदी में:\n\nSame circle में equal chords equal central angles subtend करते हैं।
हिंदी में Step-by-Step Solution:\n\n
AB = CD. Therefore the central angles subtended by them are equal: ∠AOB = ∠COD. Given ∠AOB = 70°, so ∠COD = 70°.
✅ ∠COD = 70°.
Important Question 11
Perpendicular from Centre to a Chord
A perpendicular from centre O meets chord AB at M. If AB = 16 cm, find AM and MB.
आसान हिंदी में:\n\nCentre से chord पर खींची गई perpendicular chord को bisect करती है। इसलिए M, AB का midpoint होगा।
हिंदी में Step-by-Step Solution:\n\n
Since OM ⟂ AB and O is the centre, M bisects AB. Thus AM = MB = AB/2 = 16/2 = 8 cm.
✅ AM = 8 cm and MB = 8 cm.
Important Question 12
Midpoint of a Chord
In a circle, M is the midpoint of chord XY and O is the centre. What can you say about OM and XY?
आसान हिंदी में:\n\nChord के midpoint को centre से जोड़ने वाली line chord पर perpendicular होती है।
हिंदी में Step-by-Step Solution:\n\n
Because M is the midpoint of chord XY, the line OM is perpendicular to XY.
✅ OM ⟂ XY.
Important Question 13
Equal Chords and Distance from Centre
Two chords of the same circle are 12 cm and 12 cm long. Compare their distances from the centre.
आसान हिंदी में:\n\nEqual chords same circle में centre से equal distance पर होते हैं।
हिंदी में Step-by-Step Solution:\n\n
The chords have equal lengths, so by the chord-distance property they are equidistant from the centre.
✅ Their distances from the centre are equal.
Important Question 14
Longer Chord, Closer to Centre
Two chords of a circle are 18 cm and 12 cm long. Which chord is closer to the centre?
आसान हिंदी में:\n\nSame circle में longer chord centre के closer होता है।
हिंदी में Step-by-Step Solution:\n\n
18 cm is the longer chord. Therefore it lies closer to the centre than the 12 cm chord.
✅ The 18 cm chord is closer to the centre.
Important Question 15
Distance from Centre to Chord
A circle has radius 13 cm. A chord is 24 cm long. Find the perpendicular distance from the centre to the chord.
आसान हिंदी में:\n\nCentre से chord पर perpendicular डालने पर chord दो बराबर parts में divide होगा। Half chord = 12 cm और radius = 13 cm मिलेगा।
हिंदी में Step-by-Step Solution:\n\n
Let M be the midpoint of the chord. Then AM = 24/2 = 12 cm and OA = 13 cm. In right triangle OMA,
OM² + AM² = OA²
OM² + 12² = 13²
OM² = 169 - 144 = 25
OM = 5 cm.
✅ Distance from centre to the chord = 5 cm.
D. Arcs and Angles in a Circle
Important Question 16
Minor and Major Arc
Two points A and B divide a circle into two arcs. One arc is shorter and the other is longer. What are these arcs called?
आसान हिंदी में:\n\nShorter arc को minor arc और longer arc को major arc कहते हैं।
हिंदी में Step-by-Step Solution:\n\n
The shorter part of the circle is the minor arc and the longer part is the major arc.
✅ Shorter = minor arc; longer = major arc.
Important Question 17
Central Angle and Angle at the Circle
An arc of a circle subtends an angle of 120° at the centre. What angle does the same arc subtend at a point on the remaining circle?
आसान हिंदी में:\n\nSame arc का central angle, circle पर बने angle का double होता है। इसलिए circle पर बना angle = 120° ÷ 2।
हिंदी में Step-by-Step Solution:\n\n
Central angle = 2 × angle at a point on the circle. So angle at the circle = 120°/2 = 60°.
✅ 60°.
Important Question 18
Angle in a Semicircle
AB is a diameter of a circle and C is any point on the circle. Find ∠ACB.
आसान हिंदी में:\n\nDiameter द्वारा circle पर subtended angle हमेशा 90° होता है।
हिंदी में Step-by-Step Solution:\n\n
Since AB is a diameter, the angle subtended by AB at any point C on the circle is a right angle.
✅ ∠ACB = 90°.
Important Question 19
Finding a Central Angle
An angle subtended by an arc at a point P on the circle is 35°. Find the central angle subtended by the same arc.
आसान हिंदी में:\n\nCentral angle = 2 × angle at a point on the circle।
हिंदी में Step-by-Step Solution:\n\n
Central angle = 2 × 35° = 70°.
✅ 70°.
Important Question 20
Same Arc, Same Angle
Angles APB and AQB stand on the same arc AB of a circle. If ∠APB = 48°, find ∠AQB.
आसान हिंदी में:\n\nएक ही arc पर circle के same segment में बने angles equal होते हैं।
हिंदी में Step-by-Step Solution:\n\n
Since both angles stand on the same arc AB, they are equal. Therefore ∠AQB = 48°.
✅ ∠AQB = 48°.
E. Concyclic Points and Cyclic Quadrilaterals
Important Question 21
Concyclic Points
What does it mean for four points A, B, C and D to be concyclic?
आसान हिंदी में:\n\nअगर चारों points एक ही circle की circumference पर lie करते हैं, तो वे concyclic कहलाते हैं।
हिंदी में Step-by-Step Solution:\n\n
Four points are concyclic if all four lie on the same circle.
✅ A, B, C and D are concyclic when they lie on one common circle.
Important Question 22
Cyclic Quadrilateral
What is a cyclic quadrilateral?
आसान हिंदी में:\n\nवह quadrilateral जिसके सभी चार vertices एक ही circle पर हों, cyclic quadrilateral कहलाता है।
हिंदी में Step-by-Step Solution:\n\n
A quadrilateral is cyclic when all its vertices lie on a common circle.
✅ A quadrilateral whose four vertices lie on the same circle.
Important Question 23
Opposite Angles
In cyclic quadrilateral ABCD, ∠A = 112°. Find ∠C.
आसान हिंदी में:\n\nCyclic quadrilateral में opposite angles का sum 180° होता है।
हिंदी में Step-by-Step Solution:\n\n
∠A + ∠C = 180°. Hence ∠C = 180° - 112° = 68°.
✅ ∠C = 68°.
Important Question 24
Find the Other Opposite Angle
In cyclic quadrilateral PQRS, ∠Q = 74°. Find ∠S.
आसान हिंदी में:\n\nQ और S opposite angles हैं। उनका sum 180° होगा।
हिंदी में Step-by-Step Solution:\n\n
∠Q + ∠S = 180°. So ∠S = 180° - 74° = 106°.
✅ ∠S = 106°.
Important Question 25
Converse Test for a Cyclic Quadrilateral
A quadrilateral has one pair of opposite angles measuring 95° and 85°. Can you conclude that it is cyclic?
आसान हिंदी में:\n\nअगर किसी quadrilateral के opposite angles का sum 180° हो, तो converse theorem के अनुसार वह cyclic होता है।
हिंदी में Step-by-Step Solution:\n\n
95° + 85° = 180°. Therefore the condition for the converse is satisfied, so the quadrilateral is cyclic.
✅ Yes, it is cyclic.
F. Mixed Exam Practice
Important Question 26
Radius-Chord Right Triangle
A circle has radius 10 cm. A chord is 16 cm long. Find the perpendicular distance of the chord from the centre.
आसान हिंदी में:\n\nPerpendicular from centre chord को bisect करती है। Half chord = 8 cm. अब right triangle में Pythagoras लगाएँ।
हिंदी में Step-by-Step Solution:\n\n
Half chord = 16/2 = 8 cm. If OM is the perpendicular distance and OA = 10 cm, then
OM² + 8² = 10²
OM² = 100 - 64 = 36
OM = 6 cm.
✅ Distance = 6 cm.
Important Question 27
Central Angle from an Inscribed Angle
A point C on a circle sees chord AB under an angle of 42°. Find the angle subtended by chord AB at the centre.
आसान हिंदी में:\n\nCentral angle same arc के angle at the circle का double होता है।
हिंदी में Step-by-Step Solution:\n\n
Central angle = 2 × 42° = 84°.
✅ 84°.
Important Question 28
Semicircle Application
AB is a diameter and AC = BC in a circle. If ∠CAB = 35°, find ∠ABC and ∠ACB.
आसान हिंदी में:\n\nAB diameter है, इसलिए ∠ACB = 90°. Triangle के angles का sum 180° है।
हिंदी में Step-by-Step Solution:\n\n
Since AB is a diameter, ∠ACB = 90°. Then ∠ABC = 180° - 90° - 35° = 55°.
✅ ∠ABC = 55° and ∠ACB = 90°.
Important Question 29
Cyclic Quadrilateral Calculation
In cyclic quadrilateral ABCD, ∠A = 2x° and ∠C = (x + 30)°. Find x and both angles.
आसान हिंदी में:\n\nOpposite angles supplementary होते हैं: 2x + (x + 30) = 180.
हिंदी में Step-by-Step Solution:\n\n
2x + x + 30 = 180
3x = 150
x = 50.
Therefore ∠A = 2x = 100° and ∠C = x + 30 = 80°.
✅ x = 50°, ∠A = 100°, ∠C = 80°.
Important Question 30
Chord and Diameter Concept Check
A circle has a chord of maximum possible length. What is this chord called, and how is its length related to the radius?
आसान हिंदी में:\n\nMaximum chord वही होता है जो centre से गुजरता है, यानी diameter। Diameter radius का double होता है।
हिंदी में Step-by-Step Solution:\n\n
The maximum chord of a circle is the diameter because it passes through the centre. Its length is twice the radius.
✅ Maximum chord = diameter = 2 × radius.
⭐ Quick Revision
Circle के हर point की centre से दूरी radius के बराबर होती है।
Diameter एक chord है और इसकी length = 2 × radius.
तीन non-collinear points एक unique circumcircle determine करते हैं।
Equal chords same circle में centre से equal distance पर होते हैं।
Longer chord centre के closer होता है।
Central angle = 2 × angle subtended by the same arc at a point on the circle.
Angle in a semicircle = 90°.
Cyclic quadrilateral में opposite angles का sum 180° होता है।