Properties of Circles Part 1

Properties Of circles
Some Definitions:

Center-A point inside the circle. All points on the circle are equidistant (same distance) from the center point.
Radius-The radius is the distance from the center to any point on the circle. It is half the diameter.
Diameter-The distance across the circle. The length of any chord passing through the center. It is twice the radius.
Circumference-The circumference is the distance around the circle.
Chord-A line segment linking any two points on a circle.
Tangent-A line passing a circle and touching it at just one point.
Segment-the region enclosed by a chord and the circumference.
(Note the smaller area under the line”Segment” is the minor segment while the larger area is the Major Segment.)

Property 1:

A circle is symmetrical about every diameter. Hence any chord AB perpendicular to a diameter is bisected by the diameter.
Also, any chord bisected by a diameter is perpendicular to the diameter.
Proof:

Given a circle, centre O and a chord, AB, with a mid-point D, we are required to show that OĈB = 90°.
Join OA and OB. In triangle OAC and OBC,
OA = OB (radii of circle)
AC = BC (given)
OC is common.

Triangle OCD is congruent to triangle OBC (SSS property)

OĈA = OĈB.

Since these are adjacent angles on a straight line, OĈA = OĈB = 90°

Property 2

In equal circles or in the same circle, equal chords are equidistant from the centre. Chords which are equidistant from the centre are equal.

Proof

In the figure, triangle OAB is rotated through an angle AOA' to triangle OA'B' about O.

Since rotation preserves shape and size, AB = A'B' and OG = OH.


Done by:Ng Yee Hang(38)