Chord of a Circle Theorems

The chord of the circle subtends the angle at the centre of the circle which helps us to prove various concepts in the circle. There are various theorems based on the chord of a circle,

• Theorem 1: Equal Chords Equal Angles Theorem
• Theorem 2: Equal Angles Equal Chords Theorem (Converse of Theorem 1)
• Theorem 3: Equal Chords Equidistant from Center Theorem

Now, let’s discuss the same in the article below.

Theorem 1: Equal Chords Equal Angles Theorem

Statements: Equal chords subtends equal angles at the centre of the circle, i.e. the angle subtends by the chord are equal if the chord is equal.

Proof:

From the figure,

In ∆AOB and ∆DOC

• AB = CD …eq(i) (Given)
• OA = OD …eq(ii) (Radius of Circle)
• OB = OC …eq(iii) (Radius of Circle)

Thus, by SSS congruency conditions the Triangle ∆AOB and ∆COD are congruent.

Thus,

∠AOB = ∠DOC (By CPCT)

Thus, the theorem is verified.

Theorem 2: Equal Angles Equal Chords Theorem (Converse of Theorem 1)

Statement: Chords subtending equal angles at the centre of a circle are equal in length. This is the converse of the first theorem.

From the figure,

In ∆AOB and ∆DOC

• ∠AOB = ∠DOC …eq(i) (Given)
• OA = OD …eq(ii) (Radius of Circle)
• OB = OC …eq(iii) (Radius of Circle)

Thus, by SAS congruency conditions the Triangle ∆AOB and ∆COD are congruent.

Thus,

AB = CD (By CPCT)

Thus, the theorem is verified.

Theorem 3: Equal Chords Equidistant from Center Theorem

Statement: Equal chords are equidistant from the centre, i.e. the distance between the centre of the circle and the equal chord is always equal.

From the figure,

In ∆AOL and ∆COM

• ∠ALO = ∠CMO …eq(i) (90 degrees)
• OA = OC …eq(ii) (Radius of Circle)
• OL = OM …eq(iii) (Given)

Thus, by RHS congruency conditions the Triangle ∆AOB and ∆COD are congruent.

Thus,

AL = CM (By CPCT)…(iv)

Now, we know that the perpendicular drawn from the centre bisects the chords.

From eq(iv)

2AL = 2CM

AB = CD

Thus, the theorem is verified.

Chord of a circle is the line that joints any two points on the circumference of the circle. A circle can have various chords and the largest chord of a circle is the diameter of the circle. We can easily calculate the length of the chord using the Chord Length Formula. As the name suggests it is the formula for calculating the length of the chord in a circle in Geometry.

In this article, we will learn about the definition of the chord, theorems of the chords and the circle, explain its properties, and the formulas to calculate the length of the chord using different methods. The article also has some solved sample problems for better understanding.