Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Construct tangents to two given circles (internal and external).
• Construct a tangent to a circle from a point outside it.
• Apply the four-centre method to construct an oval.
• Apply tangents to create smooth transitions between lines and arcs.
• Explain practical applications of tangents in engineering.

Lesson summary

Advanced geometrical constructions build upon the foundational skills acquired in earlier grades. They provide the essential understanding and precision needed for accurate and efficient design work in various engineering fields. In South Africa, with a growing need for infrastructure development and local manufacturing, mastering these constructions is crucial for future engineers, architects, and designers. This week, we'll focus on constructing tangents to circles and arcs, including internal and external tangents, and introducing the concept of ovals, specifically using the four-centre method. These skills are used daily in designing everything from road layouts to machine parts.

Lesson notes

2.1 Tangents to Circles A tangent to a circle is a straight line that touches the circle at only one point (the point of tangency). At the point of tangency, the tangent line is perpendicular to the radius of the circle. Understanding this perpendicular relationship is fundamental to all tangent constructions. 2.1.1 Tangent from a Point Outside a Circle: Concept: We need to find the point on the circle where the line from the external point will be tangent. This point is where the radius is perpendicular to the tangent line.

Procedure: Given: A circle with center O and a point P outside the circle.

Join: Connect point P to the center O of the circle.

Bisect: Bisect the line PO (find its midpoint, M). This midpoint will be the center of our new circle.

Draw: Draw a circle with center M and radius MO (or MP, since M is the midpoint).

Intersection: The new circle will intersect the original circle at two points, T1 and T

2. These are the points of tangency.

Draw: Draw lines PT1 and PT

2. These are the tangents from point P to the circle.

Why it Works: The angle PTO is a right angle because it's an angle inscribed in a semicircle (the semicircle having PO as its diameter).

Therefore, PT is perpendicular to OT at T, fulfilling the definition of a tangent. 2.1.2 Common Tangents to Two Circles There are two types of common tangents: External Tangents and Internal Tangents.

External Tangent: Concept: A line that touches both circles on the same side (either above or below).

Procedure: Given: Two circles with centers O1 and O2 and radii r1 and r2 (let's assume r1 > r2).

Subtract: Draw a circle with center O1 and radius (r1 - r2).

Tangent: Draw a tangent from O2 to this smaller circle. Let the point of tangency be T

3. Parallel: Draw a line from O1 through T3, extending it to intersect the original circle with center O1 at point T

1. Parallel: Draw a line parallel to O1T1 from O2 to intersect the other original circle at T

2. Join: Join T1 and T

2. The line T1T2 is the external tangent.