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Circle Theorems - Circles
A circle is a set of points which are all a certain distance from a fixed point known as the centre. A line joining the centre of a circle
to any of the points on the circle is known as a radius. The circumference of a circle is the length of the circle. The circumference of
a circle = 2 × × the radius.
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. 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
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.
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. But x + y is the size of the angle we wanted to find.
Tangents - A tangent to a circle is a straight line which touches the circle at only one point (so it does not cross the circle- it just
touches it). A tangent to a circle forms a right angle with the circle's radius, at the point of contact of the tangent.
Also, 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.
Angle at the Centre
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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.You might have to be able to prove
OA = OX since both of these are equal to the radius of the circle. The triangle AOX is therefore isosceles and so OXA = a.…read more