Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

This page supports the lesson note with a companion video and a short classroom-ready summary.

For class groups and homework, share this lesson page so learners also get the summary, objectives, and full lesson context.

Performance objectives

• Construct tangents to a circle from a point outside the circle.
• Construct tangents to two circles of equal diameter.
• Construct tangents to two circles of unequal diameter.
• Draw an arc of a given radius tangent to two given lines.
• Draw an arc of a given radius tangent to a line and a circle.

Lesson summary

This week, we delve into advanced geometrical constructions, building upon the fundamental principles you've learned. Geometrical construction is not just an academic exercise; it's the bedrock of precise drawing and design, crucial in various fields like architecture, engineering, and manufacturing. Think about designing a new RDP house with optimal space and functionality, planning efficient irrigation systems for farms in the Karoo, or creating intricate patterns for traditional Zulu beadwork. All these rely on the principles of geometrical construction. A solid understanding of these principles will empower you to solve complex design problems accurately and efficiently.

Lesson notes

Tangents to Circles: Fundamental Principles A tangent is a straight line that touches a circle (or any curve) at only one point. This point is called the point of tangency. The radius drawn from the centre of the circle to the point of tangency is always perpendicular to the tangent line. This property is crucial for constructions involving tangents.

Construction 1: Tangent to a Circle from a Point Outside the Circle Procedure: Given: Circle with centre O, and point P outside the circle.

Join: Draw a straight line connecting point P to the centre O (PO).

Bisect: Bisect the line P

O. Let the midpoint be

M. Draw: Draw a circle with centre M and radius MO (or MP – they are equal since M is the midpoint).

Identify Tangent Points: This circle will intersect the original circle at two points. Label these points T1 and T

2. These are the points of tangency.

Draw Tangents: Draw straight lines connecting point P to T1 and P to T

2. These lines (PT1 and PT2) are the tangents to the circle from point

P. Why this works: The angle in a semicircle is a right angle. Angle OT1P and OT2P are right angles because they are angles in semicircles (diameter OP). Since the radius (OT1 and OT2) are perpendicular to the lines PT1 and PT2 at the points T1 and T2 respectively, PT1 and PT2 are tangents.