Circle - Class 9 Maths Chapter 8 Short Notes (Mind Maps)

There are 9 theorems in NCERT Class 9 Maths Chapter 9 on Circles. They cover equal chords and distances from the centre, angles subtended by arcs (including the rule that a central angle is twice the inscribed angle), the angle in a semicircle being 90°, and properties of cyclic quadrilaterals where opposite angles sum to 180°.

Theorem 4 — "The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining circle" — is the most frequently tested. It directly leads to Theorem 5 (equal angles in same segment) and Theorem 6 (angle in semicircle = 90°), so mastering Theorem 4 unlocks three theorems at once.

A cyclic quadrilateral is a four-sided polygon whose all four vertices lie on a circle. Its key property is that opposite angles are supplementary, meaning each pair of opposite angles adds up to 180°. For example, in cyclic quadrilateral ABCD: ∠A + ∠C = 180° and ∠B + ∠D = 180°. This property is directly tested in 3-mark and 5-mark CBSE problems.

Yes. The angle inscribed in a semicircle is always 90°. This is Theorem 6 in Class 9 Chapter 9. It follows from Theorem 4: since the arc is a semicircle, the central angle is 180°, and the inscribed angle is half of that — exactly 90°. This result holds for any point on the circle (other than the endpoints of the diameter).

A minor arc is the shorter arc connecting two points on a circle, while the major arc is the longer arc. If the two points divide the circle into equal halves, both arcs are called semicircles. The minor arc is associated with the minor segment (smaller region), and the major arc with the major segment (larger region).