Circle Theorems

1
Circle Theorems
2
Properties

• Circles
• Circle Theorems
• Angles Subtended on the Same Arc
• Angle in a Semi-Circle with Proof
• Tangents
• Angle at the Centre with Proof
• Alternate Segment Theorem with Proof
• Cyclic Quadrilaterals

3
Circles

• A circle is a set of points which are all a
  certain distance from a fixed point known as the
  centre (O).
• A line joining the centre of a circle to any of
  the points on the circle is known as a radius (AO
  BO).

4
Circles

• The circumference of a circle is the length of
  the circle. The circumference of a circle 2 p
  the radius.
• The red line in this diagram is called a chord.
  It divides the circle into a major segment and a
  minor segment.

5
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.

6
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.

7
Proof

• We can split the triangle in two by drawing a
  line from the centre of the circle to the point
  on the circumference our triangle touches.
• We know that each of the lines which is a radius
  of the circle (the green lines) are the same
  length. Therefore each of the two triangles is
  isosceles and has a pair of equal angles.

8
Proof

• But all of these angles together must add up to
  180, since they are the angles of the original
  big triangle.
• Therefore x y x y 180, in other words
  2(x y) 180. And so x y 90.

9
Tangents

• A tangent to a circle is a straight line which
  touches the circle at only one point
• A tangent to a circle forms a right angle with
  the circle's radius, at the point of contact of
  the tangent.

10
Tangents

• If two tangents are drawn on a circle and they
  cross, the lengths of the two tangents (from the
  point where they touch the circle to the point
  where they cross) will be the same.

11
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.

12
Proof

• 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.

13
Proof

• 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

14
Alternate Segment Theorem

• The alternate segment theorem shows that the red
  angles are equal to each other and the green
  angles are equal to each other.

15
Proof

• A tangent makes an angle of 90 with the radius
  of a circle, so we know that ?OAC x 90.
• The angle in a semi-circle is 90, so ?BCA 90.

16
Proof

• 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.

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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.