Chords and the Angles They Subtend
1.0Master Divisibility and the HCF Algorithm in Minutes
Discover how the linear properties of straight lines inside a circle control angular measurements at its core. Learn how to prove the geometric relationships between equal chord lengths, central angles, and perpendicular bisectors using triangle congruence rules to ace your Class 10 board exams.
2.0Introduction
While a circle looks like a simple, perfectly curved shape, its internal straight-line segments—called chords—follow strict, predictable rules. In this lesson, we use a simple bicycle wheel analogy to visualize how a chord subtends an angle at the center. We will break down the step-by-step mathematical proofs for two foundational equal-chord theorems and explore what happens when a perpendicular line is dropped from the center axle directly onto a chord.
3.0Learning Outcomes
After completing this lesson, you will be able to:
- State Euclid's Division Lemma precisely along with its required mathematical conditions.
- Identify the dividend, divisor, quotient, and remainder within the lemma's structural formula.
- Execute Euclid's Division Algorithm to calculate the HCF of large integers step-by-step.
- Solve board exam proof problems involving the odd/even structural forms of integers (e.g., 2q, 2q+1).
How do the linear properties of chords inside a circle dictate the angular measurements at its core? To explore this, we use a simple physical analogy. Imagine tying a piece of thread tightly between two points on the rim of a bicycle wheel. This taut thread acts as a geometric chord. If you draw two straight lines (radii) connecting the endpoints of this thread directly to the center axle of the wheel, the chord subtends an angle at the centre.
If you rotate the wheel mechanically, the thread moves along with it to a new position. In this new location, the thread forms a second chord. Because it is the exact same piece of thread, the length of the first chord and the second chord are perfectly identical. Intuition tells us that since their lengths didn't change, the angle they look at from the center must also remain identical. In geometry, we formalize this intuitive behavior through two foundational theorems using triangle congruence.
4.0Theorem 1: Equal Chords and Central Angles
Statement: Equal chords of a circle subtend equal angles at the centre of the circle.
The diagram shows two equal chords on a circle, joined to the center by some radii.
Since radii in a circle are equal, we know that OA, OB,and OC
OA,OB,OC and OD are all equal.
We can write this information as:
OA=OC (Radii of a circle are equal)
OB=OD (Radii of a circle are equal)
Since we are given that the chords are equal, we also know that:
AB=CD(Given)
So △OAB≡△OCD
△OAB≡△OCD by SSS.
As a result of the two triangles being congruent, we know that
∠AOB=∠COD
∠AOB=∠COD since they are corresponding angles in congruent triangles.
5.0Theorem 2: Converse of Theorem
Chords of a circle that subtend equal angles at the centre are equal.
Given — ∠ACB = ∠DCE
Prove that — Chord AB = Chord ED
Proof — In ∆ACB and ∆DCE
AC = DC = r (Radii of the same circle)
∠ ACB = ∠ DCE (equal angles given)
BC = EC = r (Radii of the same circle)
By SAS rule, △ACB ≅ △DCB
So, by CPCT AB = ED Hence Proved.
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7.0Supporting Study Materials
This study material, focusing on Chords and Circle Theorems, is designed according to the latest CBSE Class 10 Mathematics syllabus and NCERT guidelines. This guide provides clean geometric proofs, step-by-step congruence tracks, and clear visual summaries, ensuring absolute structural confidence for your school assessments and board examinations.
8.030-Second Quick Revision:
- A chord joins two points on a circle.
- A diameter is the longest chord.
- The angle formed at the centre is called the central angle.
- The angle formed on the circumference is called the inscribed angle.
- Angles subtended by the same chord in the same segment are equal.
- The angle at the centre is twice the angle at the circumference standing on the same chord.
9.0Previous Year Questions (PYQs)
Question: The angle subtended by a chord at the centre is 120°. Find the angle subtended at the circumference.
Solution: Using, ∠AOB=2∠APB
120 degree=2∠APB
∠APB=60∘
Answer: 60°
10.0Recommended Next Topics
- Angles Subtended by an Arc of a Circle (The Central Angle Theorem)
- Cyclic Quadrilaterals and Their Supplementary Opposite Angles
- Tangents to a Circle: Theorems, Proofs, and Contact Point Properties
- Solving Complex Chord Geometry Riders Using Trigonometric Ratios