The red line in the second diagram is called a chord. It
divides the circle into a major segment and a minor segment.
Theorems
Angles Subtended on the Same Arc
Angles formed from two points on the circumference are equal
to other angles, in the same arc, formed from those two points.
Angle in a SemiCircle
Angles formed by drawing lines from the ends of the diameter
of a circle to its circumference form a right angle. So c is a right
angle.
Angle with a Tangent
A tangent to a circle forms a right angle with the circle's
radius, at the point of contact of the tangent (a tangent to a circle is a line
that touches the circumference at one point only).
Angle at the Centre
The angle formed at the centre of the circle by lines
originating from two points on the circle's circumference is double the angle
formed on the circumference of the circle by lines originating from the same
points. i.e. a = 2b.
Proof
You might have to be able to prove this fact:
OA = OX since both of these are equal to the radius of the
circle. The triangle AOX is therefore isosceles and so ÐOXA = a Similarly, ÐOXB = b
Since the angles in a triangle add up to 180, we know
that ÐXOA = 180  2a Similarly, ÐBOX = 180  2b Since the angles around a point add up to
360, we have that ÐAOB = 360  ÐXOA  ÐBOX = 360  (180  2a) 
(180  2b) = 2a + 2b = 2(a + b) = 2 ÐAXB
Alternate Segment Theorem
This diagram shows the alternate segment theorem. In
short, the red angles are equal to each other and the green angles are equal to
each other.
Proof
You may have to be able to prove the alternate segment
theorem:
We use facts about related angles:
A tangent makes an angle of 90 degrees with the radius of a
circle, so we know that ÐOAC + x = 90. The angle in
a semicircle is 90, so ÐBCA = 90. The
angles in a triangle add up to 180, so ÐBCA + ÐOAC + y = 180 Therefore
90 + ÐOAC + y = 180 and so ÐOAC + y = 90 But OAC + x = 90, so ÐOAC + x = ÐOAC + y Hence x = y
Cyclic Quadrilaterals
A cyclic quadrilateral is a foursided figure in a
circle, with each vertex (corner) of the quadrilateral touching the
circumference of the circle. The opposite angles of such a quadrilateral add up
to 180 degrees.
Area of Sector and Arc Length
If the radius of the circle is r, Area of sector = pr² × A/360 Arc length = 2pr × A/360
In other words, area of
sector = area of circle × A/360 arc length = circumference of circle × A/360
Copyright © Matthew Pinkney 2003
