Lesson Notes By Weeks and Term v5 - Grade 10

Lesson Video

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Performance objectives

• Bisect a line segment.
• Bisect an angle.
• Construct a perpendicular line from a point to a line.
• Construct an equilateral triangle, square, and regular hexagon inscribed in a circle.
• Divide a line into a given number of equal parts.

Lesson summary

Basic geometrical constructions are the fundamental building blocks of Engineering Graphics and Design (EGD). They are essential skills for accurately representing and communicating technical ideas visually. This week, we'll focus on constructing specific geometrical shapes and figures using only a compass and straightedge (ruler without measurements). These constructions are vital for everything from architectural design to mechanical engineering and even fashion design. In South Africa, understanding these principles allows learners to participate in infrastructure development, manufacturing, and other industries critical for economic growth.

Lesson notes

2.1 Bisecting a Line Segment Definition: To bisect a line segment means to divide it into two equal parts.

Method: Given: A line segment A

B. Construction: Set the compass to a radius greater than half the length of AB. Place the compass point on A and draw an arc that extends above and below the line segment AB. Keeping the same compass radius, place the compass point on B and draw another arc that intersects the first arc at two points, C and D. Draw a straight line through points C and D. This line, CD, is the perpendicular bisector of A

B. The point where CD intersects AB is the midpoint of A

B. Why it works: Points C and D are equidistant from both A and

B. Therefore, they lie on the perpendicular bisector of A

B. Example: A farmer in Limpopo needs to divide a rectangular field equally between two sons. The field is 50m long (AB) and 30m wide. He can use this method to accurately find the midpoint of the 50m side and then draw a straight line to divide the field into two equal rectangular plots. 2.2 Bisecting an Angle Definition: To bisect an angle means to divide it into two equal angles.

Method: Given: An angle ∠BA

C. Construction: Place the compass point on vertex A and draw an arc that intersects both arms of the angle, AB and AC. Label these intersection points D and E. Place the compass point on D and draw an arc in the interior of the angle. Keeping the same compass radius, place the compass point on E and draw another arc that intersects the first arc at point F. Draw a straight line from vertex A through point

F. This line, AF, is the angle bisector of ∠BA

C. Why it works: This construction creates two congruent triangles, ADF and AEF, by the Side-Side-Side (SSS) congruence postulate.

Therefore, ∠DAF = ∠EA

F. Example: Imagine designing the layout of roads in a new housing development. If two roads converge at a 60-degree angle, the angle bisector would define the ideal location for a street lamp or traffic island, ensuring even lighting or equal traffic flow in both directions. 2.3 Constructing a Perpendicular Line Case 1: From a point on the line: Method: Given: A line l and a point P on the line.

Construction: Place the compass point on P and draw an arc that intersects the line l at two points, A and B. Ensure that A and B are equidistant from P. Increase the compass radius. Place the compass point on A and draw an arc above the line. Keeping the same compass radius, place the compass point on B and draw another arc that intersects the first arc at point

C. Draw a straight line through points P and

C. This line, PC, is perpendicular to line l.

Case 2: From a point off the line: Method: Given: A line l and a point P not on the line.

Construction: Place the compass point on P and draw an arc that intersects the line l at two points, A and B. Place the compass point on A and draw an arc on the side of the line l opposite point P. Keeping the same compass radius, place the compass point on B and draw another arc that intersects the first arc at point

C. Draw a straight line through points P and

C. This line, PC, is perpendicular to line l.

Why it works: These constructions rely on creating congruent triangles and properties of rhombuses or kites.

Example: Surveyors need to establish a perfect right angle when mapping out land boundaries for a new farm in KwaZulu-Natal. They can use these methods to ensure accurate corner angles. 2.4 Constructing Regular Polygons (Equilateral Triangle, Square, Hexagon) Inscribed in a Circle Equilateral Triangle: Given: A circle with centre

O. Construction: Choose any point A on the circle. Keeping the compass radius equal to the radius of the circle, place the compass point on A and draw an arc that intersects the circle at point B. Place the compass point on B and draw another arc that intersects the circle at point

C. Connect points A, B, and C to form equilateral triangle AB

C. Square: Given: A circle with centre

O. Construction: Draw a diameter through the circle (e.g., AB). Construct a perpendicular bisector of AB that passes through

O. This creates another diameter C

D. Connect points A, C, B, and D to form square ACB

D. Regular Hexagon: Given: A circle with centre

O. Construction: Choose any point A on the circle. Keeping the compass radius equal to the radius of the circle, place the compass point on A and draw an arc that intersects the circle at point B. Place the compass point on B and draw another arc that intersects the circle at point C. Continue around the circle in this manner until you have six points (A, B, C, D, E, F). Connect points A, B, C, D, E, and F to form a regular hexagon.

Why it works: These constructions are based on the properties of circles and the internal angles of regular polygons.

Example: Think about designing the patterns for traditional Zulu beadwork. The circular designs often incorporate hexagonal or triangular patterns.