Lesson Note on Circle Geometry: Tangent to a Circle for SSS 2

Class: Senior Secondary School 2 (SS2 / SSS2)
Term: 2nd Term
Week: 3
Age: 16 years
Duration: 45 minutes
Subject: General Mathematics
Curriculum Theme: Geometry
Previous Lesson: Geometrical Progression: Nth Term and Sum.
Topic: CIRCLE GEOMETRY
Subject Matter: Tangent to a circle radius perpendicular to tangent

Specific Objectives

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

Cognitive Domain:

• Define a tangent to a circle.
• State the theorem regarding the relationship between a radius and a tangent at the point of tangency.
• Solve problems involving tangents and radii of a circle.

Affective Domain:

• Appreciate the application of circle geometry in real-life situations.
• Show interest in solving geometric problems.

Psychomotor Domain:

• Accurately draw and label diagrams involving tangents and radii.
• Apply the tangent-radius theorem to calculate unknown lengths or angles.

Social Domain:

• Collaborate effectively with peers during problem-solving activities.

Reference Materials

The following resources were used in planning this lesson:

• 9 Years Basic Education Curriculum for Mathematics.
• State Unified Scheme of Work for Senior Secondary Schools.
• New General Mathematics for Senior Secondary Schools 2.

Instructional Materials

The teacher will teach this lesson with the aid of:

• Cardboard models showing circles, radii, and tangents.
• Chalkboard and chalk.
• Ruler, compass, and protractor for drawing.
• Worksheets with practice problems.

Rationale for the Lesson

This lesson helps pupils understand a fundamental property of circles involving tangents and radii. This knowledge is important for solving various geometric problems and has practical applications in fields like engineering and design, enabling pupils to better understand the world around them.

Prerequisite/Previous Knowledge

Pupils are expected to have basic knowledge of circles, their parts (radius, centre), and properties of triangles, including Pythagoras’ theorem.

Lesson Content/Board Summary

Relationship Between Tangent and Radius of a Circle

Definition of a Tangent to a Circle

A tangent to a circle is a straight line that touches the circle at exactly one point. This point is called the point of tangency or point of contact.

The Tangent-Radius Theorem

The tangent-radius theorem states that the radius drawn to the point of tangency is perpendicular to the tangent line. This means that the angle formed between the radius and the tangent at the point of contact is always 90 degrees.

Note: Perpendicular lines form a right angle (90°).

Solving Problems Involving Tangents and Radii

Problems involving tangents and radii often form right-angled triangles, which can be solved using Pythagoras’ theorem or basic trigonometry (SOH CAH TOA).

Example 1: Finding an unknown length

A tangent AB touches a circle with centre O at point A. If the radius OA = 8 cm and the length OB = 17 cm, find the length of the tangent AB.

Solution:

1. Step 1: Identify the right-angled triangle.
   Since the radius OA is perpendicular to the tangent AB at the point of tangency A, triangle OAB is a right-angled triangle with the right angle at A.
2. Step 2: State known values.
   OA (adjacent side) = 8 cm
   OB (hypotenuse) = 17 cm
   AB (opposite side) = ?
3. Step 3: Apply Pythagoras’ theorem.
   In a right-angled triangle, hypotenuse² = opposite² + adjacent²
   OB² = AB² + OA²
4. Step 4: Substitute values and solve for AB.
   17² = AB² + 8²
   289 = AB² + 64
   AB² = 289 – 64
   AB² = 225
   AB = √225
   AB = 15 cm

The length of the tangent AB is 15 cm.

Example 2: Finding an unknown angle

A tangent PQ touches a circle with centre O at point Q. If the radius OQ = 5 cm and the distance OP = 10 cm, find the angle POQ.

Solution:

1. Step 1: Identify the right-angled triangle.
   Since the radius OQ is perpendicular to the tangent PQ at Q, triangle OPQ is a right-angled triangle with the right angle at Q.
2. Step 2: State known values relative to angle POQ.
   OQ (adjacent side to angle POQ) = 5 cm
   OP (hypotenuse) = 10 cm
   Angle POQ = ?
3. Step 3: Choose the appropriate trigonometric ratio.
   We have the adjacent side and the hypotenuse, so we use Cosine (CAH: Cosine = Adjacent / Hypotenuse).
   Cos(angle POQ) = OQ / OP
4. Step 4: Substitute values and solve for angle POQ.
   Cos(angle POQ) = 5 / 10
   Cos(angle POQ) = 0.5
   Angle POQ = cos⁻¹(0.5)
   Angle POQ = 60°

The angle POQ is 60°.

Teaching Methods/Instructional Techniques

Discussion, Lecture, Demonstration, Question and Answer, Visual Aids

Instructional Procedures

Step 1: Introduction

Time: 5 minutes
Teaching Skill: Set Induction
Teacher’s Activity: The teacher greets the pupils and reviews the previous lesson on parts of a circle. The teacher then introduces the topic “Tangent to a circle” by asking pupils if they have observed how a bicycle wheel touches the ground or how a car tyre touches the road.
Pupils’ Activity: Pupils respond to the greetings, answer questions on the previous lesson, and listen attentively to the introduction.
Learning Point: Pupils recall previous knowledge and are prepared for the new topic.

Step 2: Definition of a Tangent

Time: 7 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher defines a tangent to a circle and the point of tangency, using a circular object or a cardboard model to demonstrate how a line can touch a circle at only one point. The teacher draws a tangent on the board.
Pupils’ Activity: Pupils listen, observe the demonstrations, and copy the definition into their notebooks.
Learning Point: Pupils understand what a tangent to a circle is.

Step 3: The Tangent-Radius Theorem

Time: 10 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher explains the tangent-radius theorem, stating that the radius drawn to the point of tangency is perpendicular to the tangent. The teacher uses a pre-made cardboard model or draws on the board to clearly illustrate the 90-degree angle formed. The teacher may guide pupils to sketch this relationship on paper.
Pupils’ Activity: Pupils listen, observe the demonstration, and sketch the diagram in their notebooks, noting the 90-degree angle.
Learning Point: Pupils understand the relationship between a tangent and a radius at the point of contact.

Step 4: Problem Solving (Example 1)

Time: 8 minutes
Teaching Skill: Guided Practice
Teacher’s Activity: The teacher presents the first example problem (Example 1 from the board summary) on the board and guides the pupils through each step of the solution, emphasizing the identification of the right-angled triangle and the application of Pythagoras’ theorem.
Pupils’ Activity: Pupils follow the teacher’s steps, ask questions for clarity, and copy the solution into their notebooks.
Learning Point: Pupils learn how to apply the tangent-radius theorem to solve problems involving unknown lengths.

Step 5: Problem Solving (Example 2)

Time: 7 minutes
Teaching Skill: Guided Practice
Teacher’s Activity: The teacher presents the second example problem (Example 2 from the board summary) and guides the pupils through its solution, demonstrating how to use trigonometric ratios when angles are involved.
Pupils’ Activity: Pupils participate in solving the problem, contributing ideas, and copying the solution.
Learning Point: Pupils learn how to apply the tangent-radius theorem with trigonometry to find unknown angles.

Step 6: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

1. Define a tangent to a circle.
2. State the relationship between a radius and a tangent at the point of contact.
3. A radius OA of a circle is 6 cm long. A tangent AB touches the circle at A. If OB = 10 cm, calculate the length of AB.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 7: Conclusion

Time: 3 minutes
Teaching Skill: Summarization
Teacher’s Activity: The teacher summarizes the key points of the lesson, reiterating the definition of a tangent and the perpendicular relationship between the radius and tangent. The teacher assigns homework which includes practice problems from the textbook.
Pupils’ Activity: Pupils listen to the summary and copy down the homework assignment.
Learning Point: Pupils consolidate their understanding and are given tasks to reinforce learning.

Lesson Keywords

• Tangent – A line that touches a circle at exactly one point.
• Radius – A line segment from the centre of a circle to any point on its circumference.
• Perpendicular – Forming a right angle (90 degrees).
• Point of Tangency – The single point where a tangent line touches a circle.
• Circle Geometry – The study of geometric properties of circles.

Differentiation

For pupils who grasp concepts quickly, the teacher will provide more challenging problems involving multiple geometric properties. For pupils needing extra support, the teacher will offer simplified diagrams, one-on-one guidance, and peer-assisted learning opportunities.

Note for teachers using this lesson plan

Ensure that the cardboard models are clear and large enough for all pupils to see the demonstrations effectively. Emphasize the importance of drawing clear diagrams for problem-solving. Encourage pupils to actively participate in discussions and to ask questions when they encounter difficulties.