Lesson Notes By Weeks and Term v4 - SHS 3

Lesson Video

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Performance objectives

• Define a tangent to a circle.
• State and explain the theorem for tangents from an external point.
• Prove that these tangents are equal in length.
• Solve problems involving lengths and angles of tangents.

Lesson summary

This lesson explores a fundamental theorem in circle geometry concerning tangents drawn from a single external point. Understanding this concept is crucial not just for passing examinations but also for its applications in real-world fields like engineering (designing pulley systems), architecture (creating aesthetic curves and supports), and even in navigation. In Ghana, we see circles and tangents in the design of the wheels of a "trotro", the gears in a fufu pounding machine, or the path a rope takes when tied around a cylindrical water tank. This lesson will build our spatial reasoning skills by proving the theorem and applying it to solve practical problems.

Lesson notes

Part 1: Recap of Key Definitions (5 mins)

Before we introduce the main theorem, let's refresh our memory on two important concepts. Circle: A set of all points in a plane that are at a fixed distance (the radius) from a fixed point (the centre). Tangent: A straight line that touches a circle at exactly one point. This point is called the point of tangency or point of contact. Key Property: The radius of a circle drawn to the point of tangency is always perpendicular (at 90°) to the tangent line at that point.

*(Teacher sketches this on the board for visual reinforcement)*

In the diagram, the line `l` is a tangent to the circle with centre O at point A. Therefore, the radius OA is perpendicular to the line `l`, meaning ∠OAl = 90°. Part 2: The Main Theorem and Its Proof (25 mins)

Evaluation guide

• Verify that tangents drawn from an external point to the same circle are equal when
• measured from their point of contact.
• Group discussions: In convenient groups, learners discuss the concept and prove that tangents
• drawn from an external point to the same circle are equal when measured from their point of contact.