Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Construct tangents to circles from a point outside the circle.
• Construct ellipses using the concentric circles method.
• Construct ellipses using the trammel method.
• Apply these constructions to solve practical EGD problems.

Lesson summary

This week, we delve deeper into advanced geometrical constructions, building upon the fundamental principles learned in previous weeks. We'll focus on constructing tangents to circles and arcs, constructing ellipses using various methods, and developing accurate parallel line techniques. Mastering these constructions is crucial for success in more complex EGD drawings and designs. These skills are particularly relevant in fields like architecture, civil engineering, and mechanical design. Imagine designing a new bridge in Cape Town; understanding how to accurately draw and represent curves and tangents is essential for its structural integrity and aesthetic appeal.

Lesson notes

2.1 Tangents to Circles and Arcs: A tangent is a straight line that touches a circle or arc at only one point, called the point of tangency. The tangent line is always perpendicular to the radius at the point of tangency. Understanding this perpendicular relationship is fundamental to constructing tangents correctly.

Construction 1: Tangent from a point outside the circle: Given: Circle with center O and point P outside the circle.

Join: Draw a line connecting point P to the center of the circle

O. Bisect: Bisect the line PO to find its midpoint,

M. Draw Arc: Using M as the center and MP as the radius, draw an arc that intersects the circle at points T1 and T

2. Draw Tangents: Draw straight lines from P to T1 and P to T

2. These are the tangents to the circle from point

P. Why this works: Angle PT1O and PT2O are angles in a semi-circle and are thus right angles.

Therefore, lines PT1 and PT2 are perpendicular to the radii OT1 and OT2 and are tangents to the circle.

Example: Imagine you're designing a roundabout (traffic circle). You need to ensure the roads joining the roundabout are tangent to the circular edge to ensure smooth traffic flow. Incorrect tangents will cause abrupt turns and potential accidents.

Construction 2: Tangent from a point on the circle: Given: Circle with center O and point P on the circumference of the circle.

Join: Draw a radius from the center O to the point P on the circumference.

Perpendicular: Construct a line perpendicular to the radius OP at point

P. This line is the tangent.

Why this works: By definition, a tangent is perpendicular to the radius at the point of contact. 2.2 Ellipse Constructions: An ellipse is a closed curve, a generalized form of a circle. It has two focal points (foci) and two axes: the major axis (longer axis) and the minor axis (shorter axis). We will explore three common methods of constructing ellipses.

Method 1: Trammel Method (also known as the pin and slot method): Given: Major and minor axes lengths.

Prepare Trammel: On a strip of paper (the trammel), mark three points: A, B, and C. The distance AB should equal half the length of the minor axis, and the distance AC should equal half the length of the major axis.

Set Up: Place the trammel on the drawing surface. The line representing the major axis will be the x-axis and the line representing the minor axis will be the y-axis.

Locate Foci: Securely fix pins at A and B so that they lie on the x and y axes respectively.

Trace Ellipse: Move the trammel, keeping point A always on the y-axis and point B always on the x-axis. Point C will trace the ellipse.

Why this works: The trammel maintains a constant relationship between the distances from the two fixed axes, ensuring that the traced curve satisfies the geometrical definition of an ellipse. This is very useful when designing oval race tracks or elliptical arches.

Method 2: Concentric Circles Method: Given: Major and minor axes lengths.

Draw Circles: Draw two concentric circles (circles with the same center). The larger circle has a diameter equal to the major axis, and the smaller circle has a diameter equal to the minor axis.

Divide into Sections: Divide both circles into the same number of equal sections (e.g., 12 or 16) by drawing radial lines from the center.

Project Lines: From each point on the larger circle, draw a vertical line down (parallel to the minor axis). From each corresponding point on the smaller circle, draw a horizontal line across (parallel to the major axis).

Locate Points: The intersection of the vertical and horizontal lines from corresponding points defines a point on the ellipse.

Draw Ellipse: Connect the points with a smooth curve to form the ellipse.

Why this works: This method relies on the relationship between circles and ellipses as scaled versions of each other. It provides a visual and geometric way to generate points on the ellipse.

Method 3: Rectangle Method: Given: Major and minor axis lengths.

Construct Rectangle: Construct a rectangle with length equal to the major axis and height equal to the minor axis.

Divide: Divide the half-length of the major axis and the half-length of the minor axis into the same number of equal parts.

Draw Lines: Draw lines from the end of the minor axis divisions to the end of the major axis. Then draw lines from the center to the major axis divisions.

Locate Points: Find the intersection of the lines from step

4. These points lie on the ellipse.

Draw Ellipse: Connect the points to form the ellipse.

Why this works: Similar to the concentric circles method, this breaks the ellipse into smaller more manageable portions and creates proportional relationships for each of the points.

Example: Consider the design of the "Voortrekker Monument" in Pretoria. Many of the architectural elements involve ellipses. Accurately representing these shapes is crucial for visualizing and constructing the monument. 2.3 Parallel Line Constructions: Parallel lines are lines that never intersect.