Properties of Circles Part 2

Property 1

An angle at the centre of a circle is twice any angle at the circumference subtended by the same arc.


Proof:

In the figure below, the angles are subtended by the minor arc AB.
Since OA = OD (radii of circle), a = b (base angles of isos. triangle)
But angle AôE if the exterior angle of triangle AOD
AôE = 2a
Similarly, c = d (base angles of isos. triangle)
BôE = 2c
Hence, AôB = 2a + 2c = 2(a + c) = 2 angle ADB

Property 2

Every angle at the circumference subtended by the diameter of a circle is a right angle triangle.

Proof

AôB = 2AĈB ( at centre = 2 at )
But AôB = 180°
AĈB = 90°

Property 3

Angles in the same segment of a circle are equal.

Proof


AôB = 2x1 = 2x2 (Ð at centre= 2Ðat ⊙ce)
x1 = x2
ÐAPB = ÐAQB

Done by:Ng Yee Hang(38)