Angle at Center vs Angle at Circumference (Prove) MCQs Quiz | Class 9

This Class IX Mathematics (Code 041) quiz focuses on Unit IV: Geometry, specifically the theorem stating that the angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle. Test your understanding, view your score, and download the PDF solution.

Overview: Angle at Center vs Circumference

In Class 9 Geometry, one of the most fundamental circle theorems states that the angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle. This relationship helps in solving complex geometry problems involving cyclic properties and chord lengths.

Key Concepts

  • The Theorem: If an arc subtends an angle ‘x’ at the center (O), then the angle subtended by the same arc at any point on the remaining part of the circle is ‘x/2’.
  • Semicircle Property: Since a diameter subtends 180 degrees at the center, the angle subtended by it at the circumference is 90 degrees. Thus, the angle in a semicircle is a right angle.
  • Same Segment: Angles subtended by the same arc at the circumference in the same segment are equal.
  • Reflex Angles: For a major arc, the angle at the center is reflex (greater than 180 degrees). The relationship still holds: Reflex Angle at Center = 2 * Angle at Circumference.

Quick Revision Table

Scenario Angle at Center (O) Angle at Circumference (P)
Minor Arc 2x x
Diameter (Semicircle) 180 degrees 90 degrees
Major Arc Reflex Angle (2y) Obtuse Angle (y)

Practice Questions

  1. If the angle at the center is 120 degrees, find the angle at the major arc.
  2. Prove that the angle in a major segment is acute.
  3. If an arc subtends 40 degrees at the circumference, what is the reflex angle at the center? (Hint: Calculate minor center angle first).
  4. Given a chord equal to the radius, find the angle subtended by it at the minor arc.
  5. If points A, B, C are on a circle with center O and angle AOC = 100 degrees, find angle ABC.