Subjects

Maths

Geometry & measures

Angles

Circle Theorems and Cyclic Quadrilateral Properties for Class 9 and 10

Name: Knowunity
Brand: Knowunity

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Circle Theorems and Cyclic Quadrilateral Properties for Class 9 and 10

Fatou Barrie

@fatou.bar07

2 Followers

Circle theorems and properties explained: A comprehensive guide to understanding geometric relationships in circles

This guide covers essential circle theorems and properties, including angles subtended by diameters, relationships between central and inscribed angles, and properties of cyclic quadrilaterals.
Key concepts include the angle at the center theorem, right angles in semicircles, equal angles in the same segment, and the perpendicularity of tangents to radii.
The guide also explores more advanced topics such as the alternate segment theorem and properties of tangents from external points.

24/05/2023

164

CIRCLE
THEORUMIS 1 +K
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angle at the centre of a
circle is twice the angle at the
Circumference when both
subtended by the same ar

Theorem 1: Angle at the Center

This page illustrates the first important circle theorem, which relates angles at the center and circumference of a circle.

Definition: The angle at the center of a circle is twice the angle at the circumference when both are subtended by the same arc.

This theorem is fundamental in understanding the relationship between central and inscribed angles in a circle. It provides a powerful tool for solving problems involving circular geometry and is often used in conjunction with other circle theorems.

Example: If an angle at the circumference is 30°, the corresponding angle at the center subtended by the same arc would be 60°.

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Theorem 2: Angle in a Semicircle

This page presents another crucial circle theorem that deals with angles inscribed in a semicircle.

Highlight: The angle in a semicircle is 90°, also known as a right angle.

This theorem, often referred to as the "angle in a semicircle theorem," is widely used in geometry and has numerous practical applications. It provides a reliable method for constructing right angles and is frequently used in architectural and engineering designs.

Example: Any triangle inscribed in a semicircle will always have a right angle opposite the diameter.

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Theorem 8: Alternate Segment Theorem

This page presents the alternate segment theorem, which relates angles formed by tangents and chords.

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

This theorem is particularly useful in problems involving tangent-chord angles and provides a method for relating these angles to inscribed angles in the circle.

Example: If a tangent is drawn at point P on a circle, and a chord PQ is drawn, the angle between the tangent and chord PQ will be equal to the angle in the alternate segment formed by chord PQ.

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Additional Circle Properties

This page discusses additional properties of circles, particularly focusing on relationships involving external points and tangents.

Highlight: The line joining an external point to the center of a circle bisects the angle between the tangents drawn from that external point.

This property is useful in solving problems involving external tangents to circles and has applications in various geometric constructions.

Example: If two tangents are drawn from an external point to a circle, the line joining this point to the center of the circle will bisect the angle formed by these tangents.

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Theorem 7: Perpendicular Chord Bisector

This page introduces a theorem about the relationship between a radius and a chord of a circle.

Definition: A radius bisects a chord at 90°.

This theorem provides a method for finding the center of a circle given a chord and is often used in construction and design problems involving circles.

Vocabulary: To bisect means to divide something into two equal parts or segments.

Example: If a radius is drawn perpendicular to a chord, it will divide that chord into two equal parts.

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Theorem 3: Angles in the Same Segment

This page introduces the theorem about angles in the same segment of a circle.

Definition: Angles subtended at the circumference in the same segment of a circle are equal.

This theorem is particularly useful when dealing with inscribed angles and provides a method for proving the equality of angles without knowing their exact measures. It's a key concept in solving problems related to cyclic quadrilaterals and other circular geometries.

Example: If two chords intersect on the circumference of a circle, the angles formed at this intersection point will be equal if they are in the same segment.

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Theorem 6: Equal Tangents

This page presents a theorem about tangents drawn from an external point to a circle.

Definition: Tangents to a circle from an external point to the point of contact are equal in length.

This theorem is particularly useful in problems involving external tangents to circles and has applications in various fields, including optics and engineering.

Vocabulary: Congruent means exactly equal in size and shape.

Example: If two tangents are drawn from a point outside a circle to the circle, the lengths of these tangents from the external point to their respective points of contact on the circle will be equal.

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Theorem 5: Tangent-Radius Perpendicularity

This page introduces a fundamental property of tangents to a circle.

Definition: A tangent to a circle is perpendicular to the radius drawn to the point of contact.

This theorem is essential for understanding the relationship between tangents and radii in a circle. It has numerous applications in geometry and physics, particularly in problems involving motion along circular paths.

Vocabulary: Perpendicular means at an angle of 90° to a given line or surface.

Example: If a line is tangent to a circle at point P, and a radius is drawn to point P, these two lines will always form a right angle.

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Theorem 4: Cyclic Quadrilateral Angles

This page discusses an important property of cyclic quadrilaterals, which are quadrilaterals inscribed in a circle.

Highlight: The sum of the opposite angles of a cyclic quadrilateral is 180°.

This theorem is crucial for solving problems involving cyclic quadrilaterals and is often used in conjunction with other circle theorems. It provides a powerful tool for proving that a quadrilateral is cyclic and for calculating unknown angles in such figures.

Vocabulary: A cyclic quadrilateral is a four-sided polygon whose vertices all lie on a single circle.

Example: In a cyclic quadrilateral ABCD, if angle A = 70° and angle C = 110°, we can conclude that B + D = 180°.

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Circle Theorems Introduction

This page introduces the concept of circle theorems and properties. Circle theorems are fundamental geometric principles that describe relationships between various elements of a circle, such as angles, chords, and tangents. These theorems are crucial for solving complex geometric problems and are widely studied in mathematics education from middle school through high school levels.

Highlight: Circle theorems form the foundation for understanding advanced geometric concepts and problem-solving techniques.

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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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Download in

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Knowunity is the #1 education app in five European countries

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Pupils love Knowunity

#1

In education app charts in 12 countries

950 K+

Students have uploaded notes

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iOS User

I love this app so much, I also use it daily. I recommend Knowunity to everyone!!! I went from a D to an A with it :D

Philip, iOS User

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

Lena, iOS user

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

Subjects

Maths

Geometry & measures

Angles

Circle Theorems and Cyclic Quadrilateral Properties for Class 9 and 10

Fatou Barrie

@fatou.bar07

2 Followers

Circle theorems and properties explained: A comprehensive guide to understanding geometric relationships in circles

This guide covers essential circle theorems and properties, including angles subtended by diameters, relationships between central and inscribed angles, and properties of cyclic quadrilaterals.
Key concepts include the angle at the center theorem, right angles in semicircles, equal angles in the same segment, and the perpendicularity of tangents to radii.
The guide also explores more advanced topics such as the alternate segment theorem and properties of tangents from external points.

24/05/2023

164

7/8

Maths

Theorem 1: Angle at the Center

This page illustrates the first important circle theorem, which relates angles at the center and circumference of a circle.

Definition: The angle at the center of a circle is twice the angle at the circumference when both are subtended by the same arc.

Example: If an angle at the circumference is 30°, the corresponding angle at the center subtended by the same arc would be 60°.

Theorem 2: Angle in a Semicircle

This page presents another crucial circle theorem that deals with angles inscribed in a semicircle.

Highlight: The angle in a semicircle is 90°, also known as a right angle.

Example: Any triangle inscribed in a semicircle will always have a right angle opposite the diameter.

Theorem 8: Alternate Segment Theorem

This page presents the alternate segment theorem, which relates angles formed by tangents and chords.

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

This theorem is particularly useful in problems involving tangent-chord angles and provides a method for relating these angles to inscribed angles in the circle.

Example: If a tangent is drawn at point P on a circle, and a chord PQ is drawn, the angle between the tangent and chord PQ will be equal to the angle in the alternate segment formed by chord PQ.

Additional Circle Properties

This page discusses additional properties of circles, particularly focusing on relationships involving external points and tangents.

Highlight: The line joining an external point to the center of a circle bisects the angle between the tangents drawn from that external point.

This property is useful in solving problems involving external tangents to circles and has applications in various geometric constructions.

Example: If two tangents are drawn from an external point to a circle, the line joining this point to the center of the circle will bisect the angle formed by these tangents.

Theorem 7: Perpendicular Chord Bisector

This page introduces a theorem about the relationship between a radius and a chord of a circle.

Definition: A radius bisects a chord at 90°.

This theorem provides a method for finding the center of a circle given a chord and is often used in construction and design problems involving circles.

Vocabulary: To bisect means to divide something into two equal parts or segments.

Example: If a radius is drawn perpendicular to a chord, it will divide that chord into two equal parts.

Theorem 3: Angles in the Same Segment

This page introduces the theorem about angles in the same segment of a circle.

Definition: Angles subtended at the circumference in the same segment of a circle are equal.

Example: If two chords intersect on the circumference of a circle, the angles formed at this intersection point will be equal if they are in the same segment.

Theorem 6: Equal Tangents

This page presents a theorem about tangents drawn from an external point to a circle.

Definition: Tangents to a circle from an external point to the point of contact are equal in length.

This theorem is particularly useful in problems involving external tangents to circles and has applications in various fields, including optics and engineering.

Vocabulary: Congruent means exactly equal in size and shape.

Example: If two tangents are drawn from a point outside a circle to the circle, the lengths of these tangents from the external point to their respective points of contact on the circle will be equal.

Theorem 5: Tangent-Radius Perpendicularity

This page introduces a fundamental property of tangents to a circle.

Definition: A tangent to a circle is perpendicular to the radius drawn to the point of contact.

Vocabulary: Perpendicular means at an angle of 90° to a given line or surface.

Example: If a line is tangent to a circle at point P, and a radius is drawn to point P, these two lines will always form a right angle.

Theorem 4: Cyclic Quadrilateral Angles

This page discusses an important property of cyclic quadrilaterals, which are quadrilaterals inscribed in a circle.

Highlight: The sum of the opposite angles of a cyclic quadrilateral is 180°.

Vocabulary: A cyclic quadrilateral is a four-sided polygon whose vertices all lie on a single circle.

Example: In a cyclic quadrilateral ABCD, if angle A = 70° and angle C = 110°, we can conclude that B + D = 180°.

Circle Theorems Introduction

Highlight: Circle theorems form the foundation for understanding advanced geometric concepts and problem-solving techniques.

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.

Download in

Google Play

Download in

App Store

4.9+

Average app rating

13 M

Pupils love Knowunity

#1

In education app charts in 12 countries

950 K+

Students have uploaded notes

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

iOS User

I love this app so much, I also use it daily. I recommend Knowunity to everyone!!! I went from a D to an A with it :D

Philip, iOS User

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

Lena, iOS user