Welcome to your ultimate and comprehensive study guide for Chapter 9: Chords of a Circle. Geometry plays a foundational role in logical reasoning and spatial understanding. This chapter explores basic circular definitions, the unique circle passing through three non-collinear points, and fundamental theorems regarding perpendiculars, chords, and their distances from the centre.
1. Basic Concepts of a Circle
A circle is the locus of a moving point in a plane that remains equidistant from a fixed point called the centre.
- Radius & Diameter: The constant distance from the centre to the circumference is the radius, while a chord passing directly through the centre is the diameter.
- Chord & Arc: A line segment joining any two points on the circumference is a chord. An arc is any continuous portion of the circumference.
- Segment & Sector: A segment is bounded by an arc and a chord, whereas a sector is bounded by two radii and an intercepted arc.
2. Key Theorems on Chords and Centres
- Theorem 1: One and only one circle can pass through three non-collinear points. This forms the geometric basis for constructing circumcircles of triangles.
- Theorem 2 & 3: A straight line drawn from the centre of a circle to bisect a chord (which is not a diameter) is perpendicular to the chord. Conversely, a perpendicular drawn from the centre of a circle onto a chord bisects that chord.
- Theorem 4 & 5: If two chords of a circle are congruent, they are equidistant from the centre. Conversely, two chords of a circle that are equidistant from the centre are congruent. Furthermore, the diameter is proven to be the largest possible chord in any circle.
3. Step-by-Step Example: Finding Chord Length
Example: If the length of chord AB is 8 cm and its perpendicular distance from the center is 3 cm, find the diameter of the circle.
Step-by-Step Solution:
- Step 1: Apply the perpendicular bisector property: The perpendicular from the center bisects the chord. Therefore, half the chord length (AM) = 8 / 2 = 4 cm.
- Step 2: Form a right-angled triangle: The radius (OA), half-chord (AM), and distance from center (OM) form a right-angled triangle OMA where OA is the hypotenuse.
- Step 3: Use Pythagoras’ Theorem:
radius2 = (distance)2 + (half-chord)2
r2 = 32 + 42 = 9 + 16 = 25. - Step 4: Calculate radius and diameter:
Radius (r) = √25 = 5 cm.
Diameter = 2 × r = 2 × 5 = 10 cm.
Essential Conceptual Review Questions
Q1: Why can only one circle pass through three non-collinear points?
Answer: Non-collinear points do not lie on a single straight line. Their perpendicular bisectors intersect at exactly one unique point, which serves as the unique center equidistant from all three points, defining one and only one circumcircle.
Q2: Is the diameter always the longest chord in a circle?
Answer: Yes. By triangle inequality, any chord passing through the center achieves the maximum possible distance between two points on the circumference, making the diameter the largest chord.
⭕ Perpendicular from the Center to a Chord & Equidistant Chords
A chord is any line segment joining two distinct points on the circumference of a circle. The longest chord of a circle passes through its center and is equal to the diameter (d = 2r).
- Bisection of a Chord by a Perpendicular from the Center: A straight line drawn from the center of a circle perpendicular to a chord (that is not a diameter) bisects the chord into two equal halves.
- Pythagorean Chord Calculation: If a chord of length
2ais at a perpendicular distancedfrom the center of a circle of radiusr, they satisfy the right-triangle relationr² = a² + d². For example, ifr = 10 cmandd = 6 cm, the half-chord isa = √(100 - 36) = 8 cm, so the full chord length is16 cm. - Equidistant Chords Theorem: Two chords of a circle (or of congruent circles) are congruent if and only if they are equidistant from the center of the circle.