Mathematics - Senior Secondary 2 - Chord properties

Chord properties

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TERM: 2ND TERM

WEEK: 3

Class: Senior Secondary School 2
Age: 16 years
Duration: 40 minutes for 5 periods
Subject: Mathematics
Topic: Chord Properties
Focus: Angles Suspended by a Chord, Angles Subtended by Chord at the Centre, Perpendicular Bisectors of Chords, Angles in Alternate Segment, Cyclic Quadrilaterals.

SPECIFIC OBJECTIVES:

By the end of the lesson, students should be able to:

  1. Understand and calculate angles suspended by a chord in a circle.
  2. Demonstrate the properties of angles subtended by a chord at the centre of a circle.
  3. Understand and apply the concept of perpendicular bisectors of chords.
  4. Solve problems involving angles in alternate segments.
  5. Understand the properties of cyclic quadrilaterals and solve related problems.

INSTRUCTIONAL TECHNIQUES:

  • Question and Answer
  • Guided Demonstration
  • Discussion
  • Hands-on Model Construction
  • Proof and Deduction
  • Practical Problem-Solving

INSTRUCTIONAL MATERIALS:

  • Cardboard
  • Cardboard showing chords and segments of a circle
  • Markers and rulers
  • Compass and protractor for drawing circles
  • Worksheets with practice questions

 

PERIOD 1 & 2: Introduction to Chord Properties

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Introduce the concept of a chord in a circle and explain the angle subtended by a chord at the centre. Use diagrams to show examples of angles formed.

Students listen and ask questions for clarification.

Step 2 - Chord and Angle Suspended

Discuss angles suspended by a chord in a circle, focusing on how the angle relates to the arc of the circle.

Students observe the diagram and take notes.

Step 3 - Demonstration

Draw a circle on the board and demonstrate an angle formed by a chord and the corresponding arc. Show how the angle changes as the chord is shifted.

Students sketch similar diagrams and attempt to create their own angles with rulers.

Step 4 - Application

Introduce real-life situations where this property of chords is used (e.g., in engineering or navigation).

Students engage in a discussion and give examples of real-life applications.

NOTE ON BOARD:

  • Angles suspended by a chord in a circle are always subtended by the arc of the circle.
  • The angle subtended by a chord at the centre of the circle is twice the angle subtended at any point on the circumference.
  • Cyclic quadrilaterals are formed by four points on the circumference of a circle.

EVALUATION (5 Exercises):

  1. What is the angle subtended by a chord at the centre of the circle?
  2. Define a chord in a circle.
  3. What is the relationship between the angle subtended at the centre and the angle at the circumference?
  4. Give an example of a real-life application of angles formed by chords.
  5. How would you identify a cyclic quadrilateral?

CLASSWORK (5 Questions):

  1. What is the angle subtended by the chord AB at the centre of the circle if it measures 40°?
  2. Draw a circle and mark a chord. What is the angle subtended by the chord at the centre?
  3. If the angle subtended by a chord at the centre is 100°, what is the angle at the circumference?
  4. Explain why angles suspended by a chord are important in geometry.
  5. Describe the relationship between a chord and the arc it subtends.

ASSIGNMENT (5 Tasks):

  1. Draw a circle with a chord and measure the angle subtended at the centre.
  2. Research how chord properties are used in structural engineering.
  3. Prove that angles in the same segment of a circle are equal.
  4. Write a brief explanation of cyclic quadrilaterals and their properties.
  5. Create a model using cardboard to show angles subtended by chords in a circle.

 

PERIOD 3 & 4: Perpendicular Bisectors and Angles in Alternate Segments

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Introduce the concept of perpendicular bisectors of chords. Explain how the perpendicular bisector of a chord passes through the centre of the circle.

Students listen and observe the teacher’s demonstration.

Step 2 - Perpendicular Bisector Model

Lead the students in constructing a model using cardboard to show the perpendicular bisector of a chord.

Students work in pairs to create the model with cardboard and observe the bisector.

Step 3 - Angles in Alternate Segment

Explain the concept of angles in alternate segments. Show how an angle in one segment is equal to an angle in the alternate segment.

Students engage in a demonstration and take notes.

Step 4 - Guided Practice

Work with the class to solve problems involving perpendicular bisectors and angles in alternate segments.

Students practice solving problems individually or in groups.

NOTE ON BOARD:

  • Perpendicular bisector of a chord passes through the centre of the circle.
  • Angles in alternate segments are always equal.

EVALUATION (5 Exercises):

  1. What is the perpendicular bisector of a chord?
  2. State the property of angles in alternate segments.
  3. Prove that the angle subtended by a chord at the centre is double the angle at the circumference.
  4. Describe the steps to construct a perpendicular bisector of a chord.
  5. How does the perpendicular bisector relate to the circle’s centre?

CLASSWORK (5 Questions):

  1. Construct a perpendicular bisector for the chord AB in a given circle.
  2. What happens when you draw the perpendicular bisector of a chord?
  3. In a circle, if angle ACB = 40° and angle DCB = 40°, what can you conclude about the chord AB and CD?
  4. Solve for the angle formed by a chord and the tangent at the point of contact.
  5. What is the angle formed in the alternate segment if angle ACB = 50°?

ASSIGNMENT (5 Tasks):

  1. Draw a circle and construct a perpendicular bisector of a chord.
  2. Prove that the opposite angles of a cyclic quadrilateral are supplementary.
  3. Create a model to show the relationship between angles in alternate segments.
  4. Research real-world applications of perpendicular bisectors in architecture.
  5. Solve a practical problem involving angles in alternate segments.

 

PERIOD 5: Cyclic Quadrilaterals

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Introduce cyclic quadrilaterals, explaining their properties. Discuss how these quadrilaterals are formed by four points on the circumference of a circle.

Students listen and ask questions for clarification.

Step 2 - Proof of Cyclic Quadrilaterals

Lead the class in a formal proof of the property of cyclic quadrilaterals (opposite angles are supplementary).

Students follow the proof and participate in the discussion.

Step 3 - Practical Problem Solving

Guide students in solving practical problems involving cyclic quadrilaterals.

Students work individually or in pairs to solve problems.

Step 4 - Real-life Applications

Discuss real-life applications of cyclic quadrilaterals, such as in the design of wheels or round tables.

Students engage in a group discussion about the real-life applications.

NOTE ON BOARD:

  • A cyclic quadrilateral is a quadrilateral whose vertices lie on the circumference of a circle.
  • Opposite angles of a cyclic quadrilateral are supplementary (sum to 180°).

EVALUATION (5 Exercises):

  1. Define a cyclic quadrilateral.
  2. State the property of opposite angles of a cyclic quadrilateral.
  3. Prove that the opposite angles of a cyclic quadrilateral are supplementary.
  4. Solve a problem involving a cyclic quadrilateral.
  5. Explain how cyclic quadrilaterals are used in everyday life.

CLASSWORK (5 Questions):

  1. If angle A + angle C = 180° in a cyclic quadrilateral, what is the relationship between the angles?
  2. Solve for the missing angle in a cyclic quadrilateral where the other three angles are given.
  3. Draw a cyclic quadrilateral and label its properties.
  4. Prove that the opposite angles of a cyclic quadrilateral are supplementary.
  5. Identify cyclic quadrilaterals in real-life objects.

ASSIGNMENT (5 Tasks):

  1. Research the role of cyclic quadrilaterals in astronomy.
  2. Write a proof showing the sum of angles in a cyclic quadrilateral is 360°.
  3. Solve a real-life problem involving cyclic quadrilaterals.
  4. Create a diagram of a cyclic quadrilateral and label the properties.
  5. Describe an engineering application of cyclic quadrilaterals.

 

SUMMARY AND CONCLUSION:

  1. Recap all the properties discussed: angles suspended by chords, perpendicular bisectors, alternate segments, and cyclic quadrilaterals.
  2. Reinforce the practical applications of these properties in real-life scenarios.
  3. Conclude with a brief discussion on how these geometric properties are foundational to higher-level mathematics and design.