What are the 7 Circle Theorems? (List with Diagrams & Examples)
The concept of circle theorems plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. These theorems show how angles, chords, tangents, and segments work inside a circle. Mastering circle theorems is useful for quick answering of geometry questions, especially in Class 9 and Class 10 exams. Let’s explore their meanings, key formulas, common mistakes, and simple tricks to remember them for your exams.
What Is Circle Theorem?
A circle theorem is a rule that describes the relationship between angles, lines, and points in a circle. You’ll find this concept applied in topics like chord geometry, tangent properties, and cyclic quadrilaterals. Knowing circle theorems lets you find unknown angles or prove shapes in a circle easily, without using a protractor.
List of Key Circle Theorems
- Angle at the Centre Theorem: The angle at the centre is twice the angle at the circumference subtended by the same arc.
- Angles in the Same Segment: Angles subtended by the same chord in the same segment are equal.
- Angle in a Semicircle: An angle formed in a semicircle is 90° (right angle).
- Perpendicular from Centre to Chord: The perpendicular from the centre to a chord bisects the chord.
- Equal Chords Theorem: Equal chords are equidistant from the centre.
- Tangent-Radius Theorem: The tangent to a circle is perpendicular to the radius at the point of contact.
- Cyclic Quadrilateral Theorem: Opposite angles of a cyclic quadrilateral sum to 180°.
- Alternate Segment Theorem: The angle between a tangent and a chord is equal to the angle in the alternate segment.
- Tangent-Secant Theorem: The square of the tangent equals the product of the external and the entire length of the secant.
Key Formula for Circle Theorems
Here’s the standard formula for the angle at the centre theorem: \( \text{Angle at the centre} = 2 \times \text{Angle at the circumference} \)
For the tangent-secant theorem: \( PT^2 = PQ \times PR \) where PT is the tangent, and PQ, PR are parts of the secant.
Step-by-Step Illustration
- You are given a circle with centre O. Chord AB subtends angle ACB at the circumference. Given ∠ACB = 40°.
- By the angle at centre theorem, the angle subtended by AB at the centre (∠AOB) is:
∠AOB = 2 × ∠ACB = 2 × 40° = 80°
Speed Trick or Vedic Shortcut
Here’s a quick shortcut for circle theorem questions: If two chords are the same length, they are the same distance from the centre—no need to calculate distance every time; just check lengths.
Example Trick: When you see two equal chords, immediately declare their distances from the centre are equal—saving time during exams.
Try These Yourself
- State the angle made by a diameter at the edge of a circle.
- Given two equal chords, what is their distance from the centre?
- If a tangent meets a radius at a point, what angle do they form?
- A cyclic quadrilateral has one angle 110°. What is its opposite angle?
Frequent Errors and Misunderstandings
- Confusing ‘angle at centre’ with ‘angle at circumference’—remember, centre angle is always larger!
- Forgetting that the tangent makes 90° with the radius, not with the chord.
- Mistaking a cyclic quadrilateral’s opposite angles as equal (they sum to 180°, not always equal).
Relation to Other Concepts
The idea of circle theorems connects closely with cyclic quadrilaterals and tangent properties. Mastering these theorems also helps in trigonometry, area calculations, and proofs involving parts of a circle. Understanding circle theorems is especially helpful before learning about angles in polygons in higher geometry.
Classroom Tip
A quick way to remember key circle theorems is using the phrase: “Centre is twice, Tangent is right, Same segment means equal, Opposite in cyclic adds to 180°.” Vedantu’s teachers often use diagrams and colour codes in class to help students visualize and retain circle theorems better.
We explored circle theorems—from definitions, formulas, examples, and common mistakes to tricks and concept links. Continue practicing with Vedantu to become confident in solving all types of circle geometry problems and ace your exams!
FAQs on Circle Theorem: Rules, Diagrams & Exam Questions Explained
1. What are circle theorems?
Circle theorems are geometrical rules describing relationships between angles, chords, tangents, and arcs within a circle. Understanding these theorems helps solve various geometry problems efficiently. Key concepts include angles subtended by arcs, angles in the same segment, cyclic quadrilaterals, and properties of tangents.
2. How many circle theorems are there?
The number of circle theorems varies depending on the textbook or curriculum. While some sources list seven key theorems, others may include up to fourteen, incorporating variations and extensions of the core concepts. It's best to refer to your specific textbook or syllabus for a definitive list.
3. What is the alternate segment theorem?
The alternate segment theorem states that the angle between a tangent and a chord through the point of contact is equal to the angle subtended by the chord in the alternate segment of the circle. This means the angle formed outside the circle by the tangent and chord is equal to the angle formed inside the circle in the opposite segment.
4. What is the theorem about angles at the center and circumference?
The angle subtended by an arc at the center of a circle is twice the angle subtended by the same arc at any point on the circumference. This relationship is crucial for solving problems involving angles within a circle.
5. Explain the theorem concerning angles in the same segment.
Angles subtended by the same arc in the same segment of a circle are equal. This means if you have two angles subtended by the same arc within the same region of the circle, those angles will always have the same measure.
6. What is the angle in a semicircle theorem?
The angle subtended by a diameter at any point on the circumference of a circle is always a right angle (90°). This theorem is frequently used in solving problems involving right-angled triangles inscribed within a circle.
7. What are the properties of cyclic quadrilaterals related to circle theorems?
A cyclic quadrilateral is a quadrilateral whose vertices all lie on the circumference of a circle. In a cyclic quadrilateral, the sum of opposite angles is always 180°. This property is essential for solving problems involving cyclic quadrilaterals inscribed in circles.
8. How can I easily remember circle theorems?
To effectively remember circle theorems, use a combination of techniques: * Create flashcards with diagrams and concise explanations. * Use mnemonics or rhymes to associate theorems with their properties. * Practice applying the theorems to a variety of problems. * Draw diagrams and label key parts to help visualize each theorem.
9. How are circle theorems applied in real-life situations?
Circle theorems have practical applications in various fields, including engineering (designing circular structures), architecture (creating curved designs), and surveying (measuring distances and angles). They are fundamental to understanding spatial relationships in circular geometries.
10. What are some common mistakes students make when using circle theorems?
Common errors include: * Misapplying theorems to inappropriate situations. * Confusing angles subtended at the center and circumference. * Incorrectly identifying cyclic quadrilaterals. * Failing to recognize when multiple theorems need to be applied to solve a complex problem.
11. Can two different circle theorems apply to the same diagram?
Yes, often problems require using multiple circle theorems together. For example, you might use the angle in a semicircle theorem to establish a right angle and then apply the Pythagorean theorem or trigonometric ratios to find missing lengths or angles. This often happens in proofs and more challenging problems.
12. What resources can help me practice using circle theorems?
Numerous resources are available to assist with practicing circle theorems. These include: * **Textbooks:** Your mathematics textbook will contain numerous examples and exercises. * **Online resources:** Many websites and educational platforms offer practice problems and interactive learning modules. * **Worksheet:** Look for worksheets that specifically target circle theorem problems. * **Past papers:** Working through past examination papers helps in understanding the types of questions and applying the theorems in different scenarios.
