CBSE Class 9 Maths: Circles Ex 10.2 – Chords and Angles at the Centre

Understanding Chords and Angles at the Centre (Theorems 10.1 & 10.2)

In 'Circles Ex 10.2', we delve into two cornerstone theorems that establish a direct relationship between chords and the angles they subtend at the centre of a circle. These theorems are fundamental for solving many geometry problems related to circles and are often proven using the congruence of triangles. Learning these is key to mastering 'circles ex 10 2 class 9 ncert'.

Theorem 10.1: Equal chords of a circle subtend equal angles at the centre.
This theorem states that if you have two chords within the same circle (or congruent circles) that are equal in length, then the angles formed by joining the endpoints of each chord to the centre of the circle will be identical. For instance, if chord AB and chord CD are equal in length, then the angle ∠AOB will be equal to ∠COD, where O is the centre. The proof typically involves showing that ΔAOB and ΔCOD are congruent using the SSS (Side-Side-Side) criterion, as OA = OB = OC = OD (all are radii) and AB = CD (given).

Theorem 10.2 (Converse of Theorem 10.1): If the angles subtended by the chords of a circle at the centre are equal, then the chords are equal.
This is the inverse of the first theorem. If you know that two chords subtend equal angles at the centre of a circle, then you can conclude that these two chords must have the same length. So, if ∠POQ = ∠ROS, where O is the centre and P, Q, R, S are points on the circle, then chord PQ will be equal in length to chord RS. The proof for this converse theorem uses the SAS (Side-Angle-Side) congruence criterion for ΔPOQ and ΔROS, as PO = QO = RO = SO (radii) and ∠POQ = ∠ROS (given). Both these theorems are interlinked and are frequently used together to prove properties or find unknown values in problems. A thorough understanding of their proofs helps in applying them effectively.

Key Terms for Circles

Circle

A circle is a collection of all points in a plane that are at a fixed distance (radius) from a fixed point (centre).

Centre (O)

The fixed point inside a circle from which all points on the circle are equidistant.

Radius (r)

The fixed distance from the centre to any point on the circle. All radii of the same circle are equal in length.

Chord

A line segment joining any two points on the circumference of a circle.

Subtend

When a line segment (like a chord) forms an angle at a particular point, it is said to 'subtend' that angle at that point. For example, a chord AB subtends ∠AOB at the centre.

Worked Examples: Applying Theorems 10.1 and 10.2

• Example 1: Using Theorem 10.1 (Equal chords subtend equal angles) Q: In a circle with centre O, chords AB and CD are equal in length. If ∠AOB = 70°, find the measure of ∠COD. A: Step 1: Understand the given information. We are given a circle with centre O, and two chords AB and CD. We know that AB = CD (equal chords). We are also given that the angle subtended by chord AB at the centre, ∠AOB, is 70°. Step 2: Apply Theorem 10.1. Theorem 10.1 states that equal chords of a circle subtend equal angles at the centre. Since chord AB = chord CD, the angle subtended by chord CD at the centre (∠COD) must be equal to the angle subtended by chord AB at the centre (∠AOB). Step 3: State the conclusion. Therefore, ∠COD = ∠AOB = 70°. Final answer: ∠COD = 70°.
• Example 2: Using Theorem 10.2 (Equal angles imply equal chords) Q: Two chords PQ and RS of a circle with centre O subtend angles ∠POQ = 50° and ∠ROS = 50° respectively at the centre. If the length of chord PQ is 8 cm, find the length of chord RS. A: Step 1: Understand the given information. We have a circle with centre O. Chords PQ and RS are given. We are told that the angles they subtend at the centre are equal: ∠POQ = 50° and ∠ROS = 50°. We also know the length of chord PQ is 8 cm. Step 2: Apply Theorem 10.2. Theorem 10.2 (the converse of Theorem 10.1) states that if the angles subtended by the chords of a circle at the centre are equal, then the chords are equal. Since ∠POQ = ∠ROS (both are 50°), the chords PQ and RS must be equal in length. Step 3: State the conclusion. As PQ = RS, and PQ = 8 cm, it follows that RS = 8 cm. Final answer: The length of chord RS is 8 cm.

YoLearn.ai Exam Tip: Ace Your Circle Questions!

To excel in problems from 'circles ex 10 2 class 9 ncert', remember these crucial tips:

1. Draw Clear Diagrams: Always start by drawing a neat, labelled diagram for each problem. This helps you visualize the given information and identify relationships between chords, radii, and angles. Mark the centre, radii, chords, and any given angles or lengths.
2. State the Theorem: When solving problems, explicitly mention the theorem you are using to justify your steps. For instance, write 'By Theorem 10.1, equal chords subtend equal angles at the centre' or 'By Theorem 10.2, chords subtending equal angles at the centre are equal'. This shows clarity in reasoning and earns you marks.
3. Look for Congruent Triangles: The proofs of both Theorem 10.1 and 10.2 heavily rely on triangle congruence. Often, you'll form two triangles by joining the endpoints of chords to the centre. Proving these triangles congruent (using SSS, SAS, ASA, or RHS criteria) is a common strategy to establish equality of chords or angles.
4. Read Carefully: Pay close attention to whether chords are given as equal, or if angles subtended at the centre are given as equal. This determines which theorem (Theorem 10.1 or its converse, Theorem 10.2) you need to apply.

Practice Questions with Solutions

• Q: In a circle, AB and CD are two chords such that AB = CD. If ∠COD = 80°, what is the measure of ∠AOB? A: Step 1: Identify the given information. We have a circle with equal chords AB and CD. The angle subtended by chord CD at the centre, ∠COD, is 80°. Step 2: Apply Theorem 10.1. Theorem 10.1 states that equal chords of a circle subtend equal angles at the centre. Step 3: Conclude the angle. Since AB = CD, ∠AOB must be equal to ∠COD. Final answer: Therefore, ∠AOB = 80°.
• Q: Two chords XY and ZW of a circle subtend angles of 65° and (3x - 10)° respectively at the centre. If the chords are equal in length, find the value of x. A: Step 1: Identify the given information. We have equal chords XY and ZW. The angles subtended at the centre are 65° and (3x - 10)°. Step 2: Apply Theorem 10.1. Since the chords are equal, the angles they subtend at the centre must be equal. Step 3: Set up the equation and solve for x. So, 65° = (3x - 10)°. 65 = 3x - 10 65 + 10 = 3x 75 = 3x x = 75 / 3 Final answer: x = 25.
• Q: In a circle, two chords AB and PQ are given. If ∠AOB = 45° and ∠POQ = 45° (where O is the centre), and the length of chord AB is 10 cm, what is the length of chord PQ? A: Step 1: Identify the given information. We have a circle where two chords AB and PQ subtend equal angles at the centre (∠AOB = ∠POQ = 45°). The length of chord AB is 10 cm. Step 2: Apply Theorem 10.2 (Converse of Theorem 10.1). This theorem states that if the angles subtended by the chords of a circle at the centre are equal, then the chords are equal in length. Step 3: Conclude the length. Since ∠AOB = ∠POQ, it implies that chord AB = chord PQ. Final answer: Therefore, the length of chord PQ is 10 cm.
• Q: Prove that if two chords of a circle are congruent, then their corresponding arcs are also congruent. A: Step 1: Understand the premise. We are given two congruent chords, say AB and CD, in a circle with centre O. We need to prove that arc AB is congruent to arc CD. Step 2: Relate chords to angles at the centre. By Theorem 10.1, since AB = CD, the angles they subtend at the centre are equal, i.e., ∠AOB = ∠COD. Step 3: Relate angles at the centre to arcs. In a circle, the measure of an arc is proportional to the angle it subtends at the centre. If two central angles are equal, their corresponding arcs are also equal in measure. Final answer: Since ∠AOB = ∠COD, it implies that arc AB is congruent to arc CD.

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