I’m Up and Down, and Round and Round (Class 9): Notes और MCQ Practice

आसान भाषा में

Circle plane के उन सभी points का locus (set) है जो एक fixed point (centre) से equal distance (radius) पर होते हैं। Circle में complete rotational symmetry होती है और हर diameter reflection symmetry की line होता है। इस chapter में हम chords, arcs और circumcircle की properties समझते हैं।

Important Points & Formulas

• Unique Circle: 3 non-collinear points से सिर्फ एक unique circle पास हो सकता है (collinear points से कोई circle नहीं बनता)।
• Circumcentre की Position:
• Acute-angled triangle: Triangle के inside
• Right-angled triangle: Hypotenuse के midpoint पर
• Obtuse-angled triangle: Triangle के outside
• Equal Chords & Angles: Equal length की chords centre पर equal angles subtend करती हैं (और इसका converse भी true है)।
• Perpendicular from Centre: Centre से chord पर गिराया गया perpendicular chord को bisect करता है (AM = BM).
• Distance from Centre: Equal chords centre से equidistant होती हैं।
• Longer Chord Rule: Chord जितनी लंबी होगी, centre से उसकी perpendicular distance उतनी ही कम होगी। Diameter सबसे लंबी chord है (distance = 0)।
• Chord Length Formula: Chord length = 2√(r² − d²)

(जहाँ r = radius और d = centre से chord की perpendicular distance)

• Arcs: Minor arc centre पर 180° से कम angle बनाती है, जबकि Major arc 180° से अधिक angle बनाती है।

2 Memory Tricks

1. "Right at the Mid": Right-angled triangle का circumcentre हमेशा hypotenuse के Midpoint पर टिकता है।
2. "Longer is Closer": Chord जितनी बड़ी होगी, centre के उतने ही पास (closer) होगी; diameter पर दूरी zero हो जाती है!

2 Common Board-Exam Mistakes

1. Distance Confusion: Centre से chord की distance हमेशा perpendicular distance होती है, कोई भी तिरछी (slant) line नहीं।
2. Linear Distance Ratio: अगर chord की centre से distance आधी हो जाए, तो chord की length double नहीं होती—formula 2√(r² − d²) से calculate करना ज़रूरी है।

I’m Up and Down, and Round and Round

1. Chapter in Short

This chapter introduces the geometry of circles. It explores how circular shapes appear in nature and cave art, and formally defines a circle as the locus of points equidistant from a fixed centre. The chapter covers the rotational and reflection symmetries of a circle, the number of circles that can pass through two or three points, circumcircles of triangles, and fundamental theorems connecting chord lengths, their angles subtended at the centre, and their perpendicular distances from the centre.

2. Key Concepts Explained Simply

What is a Circle?

• Centre and Radius: Imagine tying a chalk to a piece of string and holding one end fixed on a table while rotating the chalk around it. The fixed point is the centre, the length of the string is the radius, and the path drawn is the circle.
• Locus: The word "locus" simply means a collection or set of points that follow a specific rule. A circle is the locus of all points on a plane that are at a fixed distance from the centre.
• Chord and Diameter: A straight line joining any two points on a circle is a chord. If this line passes through the centre, it is the longest possible chord, known as the diameter.

Symmetries of a Circle

• Rotational Symmetry: A spinning bicycle wheel looks identical no matter how much you turn it. A circle has complete rotational symmetry (it looks the same when rotated by any angle about its centre).
• Reflection Symmetry: If you cut out a paper circle and fold it perfectly in half, the crease formed passes through the centre and is a diameter. Every diameter is a line of reflection symmetry.

How Many Circles Can Pass Through Given Points?

• Through 1 point: Infinitely many circles.
• Through 2 points (A and B): Infinitely many circles. The centres of all these circles lie on the perpendicular bisector of segment AB. The smallest such circle has AB as its diameter (radius = half of AB).
• Through 3 collinear points (points on a single straight line): No circle can pass through all three, because the perpendicular bisectors of the segments are parallel and never meet.
• Through 3 non-collinear points: Exactly one unique circle can pass through them (Theorem 1). This circle is called the circumcircle of the triangle formed by the three points, and its centre is the circumcentre.

Position of the Circumcentre of a Triangle

• Acute-angled triangle: Circumcentre lies inside the triangle.
• Right-angled triangle: Circumcentre lies exactly at the midpoint of the hypotenuse.
• Obtuse-angled triangle: Circumcentre lies outside the triangle.

Chords and Angles at the Centre

• Equal Chords Subtend Equal Angles (Theorem 2): If two chords in a circle have the same length, the angles they form with the centre are equal.
• Equal Angles Form Equal Chords (Theorem 3): If two chords make equal angles at the centre, they must have the same length.

Perpendicular from Centre to a Chord

• Midpoint to Centre (Theorem 4): A line drawn from the centre to the midpoint of a chord is perpendicular to the chord.
• Perpendicular Bisects the Chord (Theorem 5): A perpendicular dropped from the centre to any chord divides the chord into two equal halves.

Distance of Chords from the Centre

• Equal Chords are Equidistant (Theorem 6): Chords of equal length are at the exact same perpendicular distance from the centre.
• Equidistant Chords are Equal (Theorem 7): Chords that are at the same distance from the centre have equal length.
• Longer Chord is Closer to the Centre (Theorem 8): As a chord gets longer, it moves closer to the centre. The diameter has a distance of 0 from the centre and is the longest chord.

Arcs and Angles

• An arc is a continuous curved piece of the circle between two points.
• The smaller part is the minor arc (subtends an angle less than 180° at the centre).
• The larger part is the major arc (subtends an angle greater than 180° at the centre).

3. Important Definitions, Facts, and Formulas

• Circle: The locus of all points in a plane that are at an equal distance from a fixed point (the centre).
• Radius (r): The distance from the centre to any point on the circle.
• Chord: A line segment joining any two points on a circle.
• Diameter: A chord that passes through the centre of the circle; its length is 2 × r, and it is the longest chord.
• Circumcircle: The unique circle passing through all three vertices of a triangle.
• Circumcentre: The centre of the circumcircle, found at the point where the perpendicular bisectors of the sides intersect.
• Baudhāyana–Pythagoras Relation for Chords:

For a chord of length L at a perpendicular distance d from the centre in a circle of radius r:

• Half-chord length = √(r² − d²)
• Full chord length L = 2 × √(r² − d²)
• Key Facts from the NCERT Text:
• Cave paintings at Gudahandi in Odisha depict early geometric patterns like circles, triangles, and squares.
• Exactly one circle passes through three non-collinear points.
• No circle can pass through three collinear points.
• Parallel chords on opposite sides of the centre have their distance as the sum of their individual distances from the centre.

4. Solved Examples

Example 1: Finding the distance between two parallel chords

Question: Two parallel chords of lengths 6 cm and 8 cm are on opposite sides of the centre of a circle of radius 5 cm. Find the distance between the midpoints of the chords.

Solution:

1. Let the centre of the circle be O and radius r = 5 cm.
2. The perpendicular from O to a chord bisects the chord (Theorem 5).
3. For the chord of length 8 cm:

• Half-length = 8 ÷ 2 = 4 cm.
• By the Baudhāyana–Pythagoras theorem, distance from centre (d₁) is:

d₁² + 4² = 5²

d₁² + 16 = 25

d₁² = 25 − 16 = 9

d₁ = √9 = 3 cm.

1. For the chord of length 6 cm:

• Half-length = 6 ÷ 2 = 3 cm.
• Distance from centre (d₂) is:

d₂² + 3² = 5²

d₂² + 9 = 25

d₂² = 25 − 9 = 16

d₂ = √16 = 4 cm.

1. Since the chords lie on opposite sides of the centre, the distance between their midpoints is:

d₁ + d₂ = 3 cm + 4 cm = 7 cm.

Answer: 7 cm

Example 2: Finding chord length from radius and perpendicular distance

Question: Find the length of a chord in a circle where the radius is 7 cm and the perpendicular distance of the chord from the centre is 6 cm.

Solution:

1. Let the radius r = 7 cm and perpendicular distance d = 6 cm.
2. Let the half-length of the chord be x.
3. Using the right-angled triangle formed by the radius, perpendicular, and half-chord:

x² + d² = r²

x² + 6² = 7²

x² + 36 = 49

x² = 49 − 36 = 13

x = √13 cm.

1. Total chord length = 2 × x = 2√13 cm.

Answer: 2√13 cm

Example 3: Circumcentre of a right-angled triangle

Question: In a right-angled triangle ABC right-angled at B, where does the circumcentre lie, and what is the circumradius if hypotenuse AC = 10 cm?

Solution:

1. According to the properties of circumcircles, the circumcentre of a right-angled triangle lies at the midpoint of the hypotenuse.
2. The hypotenuse AC serves as a diameter of the circumcircle.
3. Circumradius = AC ÷ 2 = 10 ÷ 2 = 5 cm.

Answer: The circumcentre is at the midpoint of hypotenuse AC, and the circumradius is 5 cm.

Example 4: Proving equal chords subtend equal angles at the centre

Question: Prove that equal chords of a circle subtend equal angles at the centre of the circle (Theorem 2).

Solution:

1. Given: A circle with centre C and two chords AB and DE such that AB = DE.
2. To show: ∠ACB = ∠DCE.
3. Proof:

• In triangles CAB and CDE:
• CA = CD (radii of the same circle)
• CB = CE (radii of the same circle)
• AB = DE (given)
• Therefore, by SSS congruence criterion, ∆CAB ≅ ∆CDE.
• By corresponding parts of congruent triangles, ∠ACB = ∠DCE.

5. Common Mistakes and Quick Revision Points

• Mistake with 3 points: Thinking a circle can pass through any three points. Remember: if the three points are collinear (on one line), no circle can pass through them.
• Parallel chords distance trap: Always check if parallel chords are on the opposite sides of the centre or on the same side:
• If on opposite sides: Distance = d₁ + d₂
• If on the same side: Distance = d₂ − d₁
• Halving the chord: When applying the Baudhāyana–Pythagoras theorem, remember that the perpendicular drops to the midpoint of the chord. Use half the chord length in the equation, and double it back to get the full chord length.
• Doubling distance does not halve chord length: If the distance of a chord from the centre is doubled, the chord length does not become half. The relationship is quadratic (L = 2√(r² − d²)), not a simple direct proportion.
• Circumcentre locations:
• Acute triangle → Inside
• Right triangle → Midpoint of hypotenuse
• Obtuse triangle → Outside