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GCSE Maths > Shape and Space - Circle Theorems

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 Semi-Circle

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 OXA = a
Similarly, OXB = b


Since the angles in a triangle add up to 180, we know that XOA = 180 - 2a
Similarly, BOX = 180 - 2b
Since the angles around a point add up to 360, we have that AOB = 360 - XOA - BOX
= 360 - (180 - 2a) - (180 - 2b)
= 2a + 2b = 2(a + b) = 2 AXB

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 OAC + x = 90.
The angle in a semi-circle is 90, so BCA = 90.
The angles in a triangle add up to 180, so BCA + OAC + y = 180
Therefore 90 + OAC + y = 180 and so OAC + y = 90
But OAC + x = 90, so OAC + x = OAC + y
Hence x = y

Cyclic Quadrilaterals

A cyclic quadrilateral is a four-sided 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 = pr 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

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