Lesson Notes By Weeks and Term v4 - SHS 3

SPATIAL SENSE

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Subject: Mathematics

Class: SHS 3

Term: 2nd Term

Week: 1

Grade code: 3.3.1.LI.3

Strand code: 3

Sub-strand code: 1

Content standard code: 3.3.1.CS.1

Indicator code: 3.3.1.LI.3

Theme: GEOMETRY AROUND US

Subtheme: SPATIAL SENSE

Lesson Video

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Performance objectives

Define a tangent to a circle and its properties.
Prove that a tangent is perpendicular to the radius at the point of contact.
State and prove the Alternate Segment Theorem.
Solve problems involving tangents and angles in circles.

Lesson summary

My dear learners, today we will delve deeper into the fascinating world of circles, a shape we see everywhere, from the wheels of a tro-tro to the base of a cooking pot. We will explore the special relationship between a circle and a straight line that just touches it – a tangent. We will prove two very important theorems that are fundamental in geometry, engineering, and design. Understanding these properties allows us to solve complex problems and appreciate the beautiful logic of mathematics.

Lesson notes

Part 1: The Tangent-Radius Theorem

A. Key Definitions 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 the circumference of a circle at exactly one point. Point of Contact (or Point of Tangency): The single point where the tangent touches the circle. Radius: A line segment from the centre of a circle to any point on its circumference.

B. The Tangent-Radius Theorem The theorem states: A tangent to a circle is perpendicular to the radius drawn to the point of contact. This means that the angle between the tangent and the radius at the point of contact is always 90°.

In the diagram above, the line PQ is a tangent to the circle with centre O. The point of contact is T. The theorem states that the radius OT is perpendicular to the tangent PQ, which means ∠OTQ = ∠OTP = 90°.

Evaluation guide

Identify the tangent as perpendicular to the radius at the point of contact and verify the
Alternate Segment Theorem.
Group discussions: In convenient groups, learners discuss the concept of tangent and prove that the
tangent at the point of contact with the circumference of the circle is perpendicular to the radius.