Circles: Equal Chords–Equal Angles (Prove) MCQs Quiz | Class 9
This Class IX Mathematics (Code 041) quiz focuses on Unit IV: Geometry, specifically the topic of Circles. The questions cover theorems stating that equal chords subtend equal angles at the center and its converse. Test your understanding of these geometric proofs and properties. Submit your answers to see your score and download a detailed PDF answer sheet for revision.
Overview: Equal Chords and Angles at the Center
In the study of Geometry for Class 9, specifically within the unit on Circles, a fundamental property relates the length of chords to the angles they create at the center of the circle. This topic involves two key theorems that are converses of each other.
Theorem 1: Equal Chords Subtend Equal Angles
Statement: Equal chords of a circle subtend equal angles at the center.
Explanation: If you have two chords, say AB and CD, and their lengths are equal (AB = CD), then the angle formed by connecting points A and B to the center O (Angle AOB) is equal to the angle formed by connecting points C and D to the center O (Angle COD).
Proof Logic: This is typically proved using the SSS (Side-Side-Side) Congruence Rule. Since radii of the same circle are equal, and the chords are given as equal, the triangles formed are congruent.
Theorem 2: The Converse
Statement: If the angles subtended by the chords of a circle at the center are equal, then the chords are equal.
Explanation: This is the reverse. If Angle AOB = Angle COD, then the length of chord AB must be equal to the length of chord CD.
Proof Logic: This is proved using the SAS (Side-Angle-Side) Congruence Rule, utilizing the equal radii and the given equal included angles.
Comparison Table
| Theorem | Given Condition | Conclusion | Congruence Rule Used |
|---|---|---|---|
| Standard | Chords are equal length | Angles at center are equal | SSS (Side-Side-Side) |
| Converse | Angles at center are equal | Chords are equal length | SAS (Side-Angle-Side) |
Key Points to Remember
- Radii of the same circle are always equal. This is a crucial step in proving these theorems.
- These theorems also apply to congruent circles (circles with equal radii).
- Subtend means to stretch across or form an angle at a specific point from the endpoints of a line segment.
Extra Practice Questions
- Prove that if chords of congruent circles subtend equal angles at their centers, then the chords are equal.
- Two chords AB and AC of a circle are equal. Prove that the center of the circle lies on the angle bisector of Angle BAC.
- If two intersecting chords of a circle make equal angles with the diameter passing through their point of intersection, prove that the chords are equal.
- Given a circle with center O, if chord PQ subtends a right angle at the center and chord RS also subtends a right angle, what is the ratio of length PQ to RS?
- Can two unequal chords subtend equal angles at the center of the same circle? (Explain why not).
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