Fun Circle Theorems: Learn Triangles, Cyclic Quadrilaterals, and Chords!

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Sophia + Charlotte

28/01/2023

Maths

Maths- Circle Theorems

Subjects

Maths

Geometry and measure

Circle theorems - higher

Fun Circle Theorems: Learn Triangles, Cyclic Quadrilaterals, and Chords!

Circle theorems and angles in triangles are fundamental concepts in geometry. This summary covers key theorems, properties, and examples related to circles, cyclic quadrilaterals, and angles in various geometric configurations.

Circle theorems angles in a triangle concepts are explored through examples and formulas
Cyclic quadrilateral properties and theorems are explained with visual aids
Radius bisects chord at 90 degrees theorem is demonstrated with examples
Alternate segment theorem and its applications are discussed
Various chord properties of a circle are illustrated and explained

28/01/2023

Circle Theorems/
A
For
AB is a diameter
find the value of oc
4COB = 60°
Angles in an equilateral thonie
are 60°
AOC is an isoceles thangle a

Cyclic Quadrilaterals and Tangents

This page focuses on cyclic quadrilateral properties and tangent lines to circles. It provides several important theorems and their applications.

Definition: A cyclic quadrilateral is a four-sided figure whose vertices all lie on a circle.

The page explains that the sum of opposite angles in a cyclic quadrilateral is always 180°, which is a key property used in many geometric proofs.

Formula: For a cyclic quadrilateral ABCD, a + b + c + d = 360°, where a, b, c, and d are the angles of the quadrilateral.

This formula is essential for solving problems involving cyclic quadrilaterals and is derived from the properties of angles in a circle.

Highlight: A radius of a circle is always perpendicular to a tangent line at the point of tangency.

This property is crucial for problems involving tangent lines and is often used in conjunction with other circle theorems.

The page also mentions that tangents drawn from an external point to a circle are equal in length, which is another important property for solving circle-related problems.

View

Radius and Chord Relationships

This page delves into the relationships between radii and chords in circles, focusing on perpendicular bisectors and the alternate segment theorem.

Theorem: If a radius cuts a chord at 90°, it bisects the chord.

This theorem is fundamental in understanding the relationship between radii and chords and is often used in circle geometry problems.

Theorem: If a radius bisects a chord, it does so at a 90° angle.

This is the converse of the previous theorem and is equally important in solving circle-related problems.

Definition: The alternate segment theorem states that the angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment.

This theorem is illustrated with a diagram and a mathematical expression: 180° - $90° - x$ - $90° - x$ = 2x.

These theorems provide powerful tools for solving complex geometry problems involving circles, chords, and tangents.

View

Application of Circle Theorems

This page demonstrates the practical application of circle theorems through two detailed examples.

Example: In a circle with a tangent line, given that angle OAB = 90° and angle BOC = 52°, the problem asks to find angle x.

The solution uses the property that a radius is perpendicular to a tangent at the point of contact, and applies the theorem that angles in a triangle sum to 180°.

Example: Another problem involves finding angles x and y in a circle configuration with given angle measures.

This example utilizes multiple circle theorems, including:

Base angles in an isosceles triangle are equal
Angles on a straight line sum to 180°
Angles in a triangle sum to 180°

These examples demonstrate how to combine multiple circle theorems to solve complex geometry problems.

View

Advanced Circle Theorem Applications

The final page presents more advanced applications of circle theorems, incorporating multiple concepts in a single problem.

Example: A complex diagram is presented with various angles and segments, asking to find angle x.

This problem combines several key concepts:

Base angles in an isosceles triangle are equal
The angle at the center is twice the angle at the circumference when subtended by the same arc
Tangents from a point to a circle are equal in length

Highlight: The solution demonstrates how to break down a complex problem into smaller, manageable steps using known circle theorems.

The page emphasizes the importance of recognizing and applying multiple theorems in conjunction to solve advanced geometry problems involving circles.

Vocabulary: Subtended angle - An angle formed by two lines or planes that intersect at a point on the circumference of a circle.

Understanding this concept is crucial for applying many circle theorems correctly.

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Subjects

Maths

Geometry and measure

Circle theorems - higher

Maths

•

556

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28 Jan 2023

•

5 pages

Fun Circle Theorems: Learn Triangles, Cyclic Quadrilaterals, and Chords!

Sophia + Charlotte

@sophia.lily06

Circle theorems angles in a triangleconcepts are explored through... Show more

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Cyclic Quadrilaterals and Tangents

This page focuses on cyclic quadrilateral properties and tangent lines to circles. It provides several important theorems and their applications.

Definition: A cyclic quadrilateral is a four-sided figure whose vertices all lie on a circle.

The page explains that the sum of opposite angles in a cyclic quadrilateral is always 180°, which is a key property used in many geometric proofs.

Formula: For a cyclic quadrilateral ABCD, a + b + c + d = 360°, where a, b, c, and d are the angles of the quadrilateral.

This formula is essential for solving problems involving cyclic quadrilaterals and is derived from the properties of angles in a circle.

Highlight: A radius of a circle is always perpendicular to a tangent line at the point of tangency.

This property is crucial for problems involving tangent lines and is often used in conjunction with other circle theorems.

The page also mentions that tangents drawn from an external point to a circle are equal in length, which is another important property for solving circle-related problems.

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Radius and Chord Relationships

This page delves into the relationships between radii and chords in circles, focusing on perpendicular bisectors and the alternate segment theorem.

Theorem: If a radius cuts a chord at 90°, it bisects the chord.

This theorem is fundamental in understanding the relationship between radii and chords and is often used in circle geometry problems.

Theorem: If a radius bisects a chord, it does so at a 90° angle.

This is the converse of the previous theorem and is equally important in solving circle-related problems.

Definition: The alternate segment theorem states that the angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment.

This theorem is illustrated with a diagram and a mathematical expression: 180° - $90° - x$ - $90° - x$ = 2x.

These theorems provide powerful tools for solving complex geometry problems involving circles, chords, and tangents.

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Application of Circle Theorems

This page demonstrates the practical application of circle theorems through two detailed examples.

Example: In a circle with a tangent line, given that angle OAB = 90° and angle BOC = 52°, the problem asks to find angle x.

The solution uses the property that a radius is perpendicular to a tangent at the point of contact, and applies the theorem that angles in a triangle sum to 180°.

Example: Another problem involves finding angles x and y in a circle configuration with given angle measures.

This example utilizes multiple circle theorems, including:

Base angles in an isosceles triangle are equal
Angles on a straight line sum to 180°
Angles in a triangle sum to 180°

These examples demonstrate how to combine multiple circle theorems to solve complex geometry problems.

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Advanced Circle Theorem Applications

The final page presents more advanced applications of circle theorems, incorporating multiple concepts in a single problem.

Example: A complex diagram is presented with various angles and segments, asking to find angle x.

This problem combines several key concepts:

Base angles in an isosceles triangle are equal
The angle at the center is twice the angle at the circumference when subtended by the same arc
Tangents from a point to a circle are equal in length

Highlight: The solution demonstrates how to break down a complex problem into smaller, manageable steps using known circle theorems.

The page emphasizes the importance of recognizing and applying multiple theorems in conjunction to solve advanced geometry problems involving circles.

Vocabulary: Subtended angle - An angle formed by two lines or planes that intersect at a point on the circumference of a circle.

Understanding this concept is crucial for applying many circle theorems correctly.

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Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Circle Theorems and Angles in Triangles

This page introduces fundamental circle theorems and their applications to angles in triangles. It provides visual examples and explanations of key concepts.

Example: In a circle where AB is a diameter and angle COB is 60°, the value of angle OAC is calculated to be 30°.

The example demonstrates the relationship between angles in a circle and an equilateral triangle. It also introduces the concept of isosceles triangles within circles.

Highlight: Angles subtended from the center of a circle are twice those at the circumference when subtended by the same arc.

This principle is crucial for solving many circle-related problems and is illustrated through the given example.

Vocabulary: Isosceles triangle - A triangle with two equal sides and two equal angles.

The page uses an isosceles triangle to explain how base angles are equal and how this property can be used to solve for unknown angles in circle problems.

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Fun Circle Theorems: Learn Triangles, Cyclic Quadrilaterals, and Chords!

Cyclic Quadrilaterals and Tangents

Radius and Chord Relationships

Application of Circle Theorems

Advanced Circle Theorem Applications

Similar content

Can't find what you're looking for? Explore other subjects.

Knowunity is the #1 education app in five European countries

Knowunity has been named a featured story on Apple and has regularly topped the app store charts in the education category in Germany, Italy, Poland, Switzerland, and the United Kingdom. Join Knowunity today and help millions of students around the world.

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Still not convinced? See what other students are saying...

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The app is very simple and well designed. So far I have always found everything I was looking for :D

I love this app ❤️ I actually use it every time I study.

Fun Circle Theorems: Learn Triangles, Cyclic Quadrilaterals, and Chords!

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Cyclic Quadrilaterals and Tangents

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Radius and Chord Relationships

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Application of Circle Theorems

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Advanced Circle Theorem Applications

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Circle Theorems and Angles in Triangles

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