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9.4 - Angle Subtended by an Arc of a Circle

Interactive Audio Lesson

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Introduction to Angles and Arcs

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Teacher
Teacher

Today, we will explore the relationship between chords and arcs in circles. Can anyone tell me what happens when we have equal chords in a circle?

Student 1
Student 1

The arcs made by those chords are equal?

Teacher
Teacher

Exactly! Equal chords correspond to equal arcs. If we were to cut out the arc corresponding to one of the chords and place it over the other, they would match perfectly. This means the arcs are not just equal in length but also congruent.

Student 2
Student 2

But how do we know their lengths are equal?

Teacher
Teacher

That’s a great question. This fact is backed by the properties of chords and arcs in circles, and we’ll formalize it with some theorems shortly.

The Angle Subtended by an Arc

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Teacher
Teacher

Recall that the angle subtended by an arc at the center of a circle is defined by the chord that the arc corresponds to. For instance, in figure 9.14, if PQ is the minor arc subtending angle ∠POQ at the center O, what can be said about the angle subtended at another point A on the circle?

Student 3
Student 3

It will be half of the angle at the center?

Teacher
Teacher

Correct! This leads us to Theorem 9.7, stating that the angle at the center is twice the angle at any point on the remaining part of the circle. Let’s think of this visually—you can understand how angles relate geometrically.

Demonstrating Key Theorems

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Teacher
Teacher

Now, let’s dive deeper into Theorem 9.8 which tells us that angles in the same segment of a circle are equal. Can anyone visualize what it means to say that points subtend equal angles?

Student 4
Student 4

It means that if we have two different points but they both make the same angle with the chord, they must lie in the same segment of the circle.

Teacher
Teacher

Exactly! This is a significant concept, particularly when we work with cyclic quadrilaterals later on.

Introduction & Overview

Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.

Quick Overview

This section explains the relationship between chords and arcs in a circle, focusing on the angles they subtend at the center.

Standard

This section covers the concept of arcs created by chords in a circle, emphasizing that equal chords create equal arcs that are congruent. It introduces the definition of angles subtended by arcs at the center and at other points on the circle, exploring key theorems related to these angles.

Detailed

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Audio Book

Dive deep into the subject with an immersive audiobook experience.

Introduction to Arcs and Chords

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You have seen that the end points of a chord other than diameter of a circle cuts it into two arcs – one major and other minor. If you take two equal chords, what can you say about the size of arcs? Is one arc made by first chord equal to the corresponding arc made by another chord? In fact, they are more than just equal in length. They are congruent in the sense that if one arc is put on the other, without bending or twisting, one superimposes the other completely.

Detailed Explanation

This chunk introduces the concept of arcs created by chords in a circle. A chord that is not a diameter divides the circle into two distinct arcs – a minor arc (the smaller part) and a major arc (the larger part). When we compare two equal chords in the same circle, the arcs they subtend (i.e., the arcs that correspond to these chords) are not just equal in length; they are congruent, meaning one can be placed over the other perfectly. This is an important concept when studying circles, as it establishes a strong connection between chords and the arcs they define.

Examples & Analogies

Think of a pizza. When you cut a pizza into two slices of the same size (like equal chords), those slices not only look identical, but if you were to place one slice on top of the other, they would perfectly align. This is similar to how the arcs of equal chords in a circle behave.

Verification of Congruence

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You can verify this fact by cutting the arc, corresponding to the chord CD from the circle along CD and put it on the corresponding arc made by equal chord AB. You will find that the arc CD superimpose the arc AB completely. This shows that equal chords make congruent arcs and conversely congruent arcs make equal chords of a circle.

Detailed Explanation

In this chunk, we discuss how one can physically verify the congruence of arcs corresponding to equal chords. By cutting out the arc that corresponds to one of the equal chords and attempting to superimpose it on the arc of the other chord, one can see that they match perfectly. This hands-on activity reinforces the relationship between chords and arcs in circles: if two chords are equal, the arcs they create are congruent, and if two arcs are congruent, their corresponding chords must also be equal. This is fundamental in circle geometry.

Examples & Analogies

Imagine you have two ribbons of equal length. If you cut each ribbon in the shape of a half-circle and then try to lay one half-circle ribbon over the other, you’ll see that they fit perfectly. This illustrates the congruence of arcs created by equal chords.

Angles Subtended by Arcs

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Also, the angle subtended by an arc at the centre is defined to be angle subtended by the corresponding chord at the centre in the sense that the minor arc subtends the angle and the major arc subtends the reflex angle.

Detailed Explanation

In this section, we explore how arcs and their corresponding chords relate to angles. The angle created at the centre of the circle by a chord is defined as the angle subtended by that chord. The relationship extends to arcs, where the minor arc defines an angle at the centre and the major arc defines what is known as the reflex angle. This concept is crucial for solving problems related to angles in circles and demonstrates the interconnectedness of the various geometric properties present in circular shapes.

Examples & Analogies

Consider standing in the middle of a playground that has a circular path. If you stretch your arms to indicate the edges of a particular path section (like a chord), the angle your arms create at your shoulders represents the angle subtended by that path section. Turning to face the other direction and indicating the longer path around the circle will show you the reflex angle, encapsulating how angles relate to arcs in circles.

Theorem 9.7: Relationship of Angles

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The following theorem gives the relationship between the angles subtended by an arc at the centre and at a point on the circle. Theorem 9.7: The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

Detailed Explanation

Theorem 9.7 establishes a critical rule in circle geometry: the angle formed at the centre of a circle by an arc is always double the angle formed at any point on the remaining arc of the circle. This theorem highlights the relationship between angles and is extremely useful in solving geometry problems regarding positions and angles in circles. Understanding this theorem allows students to predict and calculate angles related to various arcs confidently.

Examples & Analogies

Imagine holding a flashlight in the center of a circular room. If you shine the light towards a section of the wall, the angle at which the light spreads at the center (where you are) is much wider than the angle created at any point you are looking at on the wall. This wider angle at your position doubles what you see as you cast light towards the wall, mirroring how angles function in a circle.

Theorem 9.8: Angles in the Same Segment

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Therefore, the angle subtended by the minor arc at O is equal to the angle subtended by the corresponding (minor) arc at the centre. Therefore, using the concept of segments, if you take any other point C on the remaining part of the circle you have ∠ POQ = 2 ∠ PCQ.

Detailed Explanation

Theorem 9.8 addresses the equality of angles subtended in the same segment of a circle. According to this theorem, if several angles are created at points on the circumference by lines connecting to the endpoints of an arc, those angles will all be equal if they lie in the same segment. This reinforces the concept that arcs and the angles formed by them can guide our understanding of circle geometry, providing a consistent framework for angle relationships.

Examples & Analogies

Think about a fan with blades. Each blade is like a line segment coming from a central point (the hub), and as you observe from different angles around the fan circle, the angle that one blade makes with lines to the edges where they connect is consistent, regardless of your point of view. This is similar to how angles in the same segment of a circle relate to one another.

Circumferential Relationships: Angle in a Semicircle

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Here ∠PAQ is an angle in the segment, which is a semicircle. Also, ∠ PAQ = 1/2 ∠POQ = 1/2 × 180° = 90°.

Detailed Explanation

This talks about the unique case of angles formed in a semicircle. Specifically, it states that if an angle is created using points on a semicircle, this angle will always equal a right angle (90°). Understanding this special case is crucial for recognizing unique properties of circles and semicircles, and takes us into deeper geometric analysis by allowing us to apply known relationships in real scenarios.

Examples & Analogies

Think of a straight edge like a ruler that you balance at the middle of a half circular track. If you were to place a little flag at either end of the radius (the ends of the straight edge), wherever you look from the curved edge of the track to any point on the half circle above (the boundary) will create a right angle with that straight edge, exhibiting the special property of angles in semicircles.

Converse of Theorem 9.8

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The converse of Theorem 9.8 is also true. It can be stated as: Theorem 9.9: If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment, the four points lie on a circle (i.e. they are concyclic).

Detailed Explanation

This is about the converse of Theorem 9.8, which introduces a criterion for establishing whether four points are concyclic (meaning they lie on the same circle). If a line segment between two points creates equal angles at two other points on the same side of that segment, then these four points form a circle. This connection is important for solving problems involving circles, as it gives a method to determine the cyclic nature of points in a plane.

Examples & Analogies

Picture four friends standing at four corners of a rectangular park with two of them holding a rope stretched diagonally across the opposite pair. If the angles they make with their respective corners to the opposite side are equal, it indicates all four could lie on the same circular path around the park, illustrating how points can form a circle based on subtended angles.

Definitions & Key Concepts

Learn essential terms and foundational ideas that form the basis of the topic.

Key Concepts

  • Congruent arcs create equal chords.

  • The angle subtended by an arc at the center is twice the angle subtended at any point on the circle.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • {'example': 'Example: If the minor arc PQ subtends an angle of 40° at point A on the circle, what is the angle subtended by the arc at the center?', 'solution': 'The angle at the center ∠POQ = 2 × 40° = 80°.'}

  • {'example': 'Example: Show that if two chords subtend equal angles at the center, then the chords are equal.', 'solution': 'Given by Theorem 9.1, the proof can be constructed using congruence criteria.'}

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎵 Rhymes Time

  • Chords equal make arcs well, together they form a telling spell.

📖 Fascinating Stories

  • Imagine two friends walking around a circle, they both walk the same distance, creating the same path—this is how equal chords relate to arcs.

🧠 Other Memory Gems

  • C-E-A: Chords Equal Arcs — remember that equal chords create equal arcs.

🎯 Super Acronyms

S.A.C.E

  • Segment Angles Chord Equivalence - reflect on how angles of segments relate to chords.

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for terms.

  • Term: Arc

    Definition:

    A part of the circumference of a circle.

  • Term: Chord

    Definition:

    A straight line connecting two points on a circle.

  • Term: Central Angle

    Definition:

    An angle whose vertex is at the center of the circle and whose sides extend to the circumference.

  • Term: Congruent

    Definition:

    Figures that are the same size and shape.