Year 10 - Maths
The angle in a semicircle is a right angle
Circle theorems
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This pupil-friendly study guide includes a unit summary, clear notes, common mistakes, and quick questions with answers.
Unit Summary
In this unit, pupils will use their developed mathematical reasoning skills to derive and use various circle theorems.
Lesson Summary
You will learn to derive and use the theorem: the angle in a semicircle is a right angle.
Key Notes
- A theorem is a statement that can be demonstrated to be true by accepted mathematical operations and arguments
- Theorems can be thought of as puzzles to solve, you are showing how to find a result
- In order to use this theorem, you may need to draw a diagram or add more information to an existing one
Vocabulary To Learn
- Subtended: An angle can be subtended by a line segment or curve. The legs of the subtended angle meet the endpoints of the line segment or curve.
- Arc: An arc of a circle is part of the circle's circumference.
- Chord: A chord is any line segment joining two points on the circumference of a circle.
- Diameter: The diameter of a circle is any line segment that starts and ends on the circumference of the circle and passes through the centre of the circle.
Common Mistakes To Avoid
- Pupils may over-generalise the circle theorem by assume that all triangles that are inscribed in a circle are right-angled.
3 Quick Questions (With Answers)
1. Solve this in steps: explain the method from this lesson and why each step matters.
A theorem is a statement that can be demonstrated to be true by accepted mathematical operations and arguments. Theorems can be thought of as puzzles to solve, you are showing how to find a result.
2. Use this key word in a maths sentence and explain it clearly. 'Subtended'
An angle can be subtended by a line segment or curve. The legs of the subtended angle meet the endpoints of the line segment or curve. Use this word when showing your method, then check your final answer is reasonable.
3. Spot and fix this common maths mistake.
Mistake: Pupils may over-generalise the circle theorem by assume that all triangles that are inscribed in a circle are right-angled. Correction: The triangle is only right-angled if one of its angles is subtended by the diameter. This also means that if the triangle is right-angled, then its hypotenuse is the diameter of the circle.
More Lessons In This Unit
- Checking and securing understanding of the parts of a circle Study Guide
- Identifying which circle theorem to use Study Guide
- Problem solving with circle theorems Study Guide
- The alternate segment theorem Study Guide
- The angle at the centre of the circle is twice the angle at any point on the circumference Study Guide
- The angles in the same segment are equal Study Guide
- The opposite angles of a cyclic quadrilateral sum to 180° Study Guide
- The perpendicular from the centre of a circle to a chord bisects the chord Study Guide
- The tangent at any point on a circle is perpendicular to the radius at that point Study Guide
- The tangents from an external point are equal in length Study Guide
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