Circle theorems - Higher
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Summary
Circle theorems
In a nutshell
Circle geometry (or circle theorems) involves looking at angle properties when lines and shapes are drawn in circles. Once you can identify and label parts of a circle, it is possible to add lines, triangles and quadrilaterals inside a circle. Circle theorems can help find missing angles.
Circle properties
You should be able to label parts of a circle.
| NAME | DESCRIPTION | ILLUSTRATION |
| Radius | A line from the centre to the edge of the circle. |  |
| Diameter | A line across the circle, going through the centre. |  |
| Circumference | The perimeter of the circle. |  |
| Arc | A section of the circumference. |  |
| Chord | A line segment joining two points on the circumference. |  |
| Sector | Part of a circle bound by an arc and two radii. |  |
| Segment | Part of a circle bound by a chord and an arc. |  |
Circle theorems
There are eight circle theorems that you should learn.
| THEOREM | DESCRIPTION | ILLUSTRATION |
| Angles in the same segment are equal. | | |
| The angle at the centre is twice the angle at the circumference. | | |
| The angle in a semicircle is . | | |
| The radius and tangent are perpendicular. | | |
| The perpendicular bisector of a chord goes through the centre. | | |
| Opposite angles in a cyclic quadrilateral add up to . | | |
| Two tangents from the same point are the same length. | | |
| The alternate segment theorem. | The angle at the tangent is equal to the angle in the alternate segment. | |
Example 1
In the circle shown, . Find , and .
is twice the angle at the circumference .
as it is the angle in a semicircle. Therefore and add up to .
is a straight line, so can be found.
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Exercises
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FAQs - Frequently Asked Questions
What is the alternate segment theorum?
The alternate segment theorum states that the angle at the tangent is the same as the angle in the alternate segment.
What is an arc?
An arc is a section of the circumference.
What is a cyclic quadrilateral?
A cyclic quadrilateral is a quadrilateral that is inside a circle, where the vertices of the quadrilateral touch the circumference of the circle. Opposite angles in a cyclic quadrilateral add up to 180 degrees.
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