Lesson Notes By Weeks and Term v5 - Grade 10

Lesson Video

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Performance objectives

• Bisect a given line segment.
• Construct an angle bisector.
• Construct a perpendicular line from a point to a line.
• Divide a given line segment into equal parts.
• Construct a regular hexagon given one side length.

Lesson summary

This week we delve into the fundamental skill of basic geometrical constructions. These constructions form the bedrock of all technical drawings and designs, from architectural blueprints for a RDP house to the intricate plans for a new bridge spanning a river. Mastery of these constructions enables accurate and precise representation, communication, and ultimately, realization of engineering ideas. Think about how a builder relies on precise angles and lines to construct a stable and aesthetically pleasing structure; geometrical constructions provide the tools to achieve that precision. Without these skills, even the simplest design projects would be impossible to execute accurately.

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. Procedure: Place the compass point at A and open the compass to a radius greater than half the length of AB (estimation is fine; accuracy will result from the intersection). Draw an arc that extends above and below the line segment AB. Without changing the compass radius, place the compass point at 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. Result: The line CD bisects the line segment AB at point

E. Therefore, AE = EB, and CD is perpendicular to A

B. Why it works: The arcs drawn from A and B with equal radii create two congruent triangles, ACD and BCD (SSS congruence).

Therefore, angles ACD and BCD are equal. Consequently, triangles ACE and BCE are also congruent (SAS congruence). This proves that AE = BE, and angle AEC = angle BEC = 90 degrees, confirming bisection and perpendicularity.

Example: Imagine a line segment representing the width of a plot of land (say, 80m) that needs to be divided equally for two siblings in a rural community. Bisecting this line accurately ensures a fair distribution of land. 2.2 Bisecting an Angle Definition: To bisect an angle means to divide it into two equal angles.

Method: Given: An angle BA

C. Procedure: Place the compass point at vertex A and draw an arc that intersects both arms of the angle, AB and AC, at points D and E, respectively. Place the compass point at D and draw an arc in the interior of the angle. Without changing the compass radius, place the compass point at E and draw another arc that intersects the first arc at point

F. Draw a straight line from vertex A through point

F. Result: The line AF bisects the angle BA

C. Therefore, angle BAF = angle CA

F. Why it works: The arcs drawn from D and E with equal radii create congruent triangles ADF and AEF (SSS congruence).

Therefore, angle DAF = angle EAF, confirming the angle bisection.

Example: A carpenter needs to construct a roof truss with a specific angle. Bisecting the angle allows for the symmetrical distribution of weight and a structurally sound design. 2.3 Constructing a Perpendicular Line from a Point to a Line Case 1: Point is on the line Given: A line AB and a point P on the line.

Procedure: Place the compass point at P and draw an arc that intersects the line AB at two points, C and D, such that PC = PD. Open the compass to a radius greater than PC. Place the compass point at C and draw an arc above (or below) the line AB. Without changing the compass radius, place the compass point at D and draw another arc that intersects the first arc at point

E. Draw a straight line through points P and

E. Result: The line PE is perpendicular to the line AB at point

P. Case 2: Point is off the line Given: A line AB and a point P off the line.

Procedure: Place the compass point at P and draw an arc that intersects the line AB at two points, C and D. Place the compass point at C and draw an arc on the opposite side of line AB from point P. Without changing the compass radius, place the compass point at D and draw another arc that intersects the first arc at point

E. Draw a straight line through points P and

E. Result: The line PE is perpendicular to the line AB at point F (where PE intersects AB).

Why it works: Similar to bisecting a line, these methods rely on creating congruent triangles to ensure a 90-degree angle.

Example: An electrician needs to install a light fixture directly above a point on the ceiling. Constructing a perpendicular line ensures the fixture hangs straight and doesn’t look crooked. 2.4 Dividing a Line Segment into Equal Parts Method (for dividing into, say, 5 equal parts): Given: A line segment A

B. Procedure: Draw a line AC at any convenient angle to AB. Using a compass, mark off five equal segments on line AC (A1, A1A2, A2A3, A3A4, A4A5). The length of these segments is arbitrary, but they should be equal. Join point A5 to point B. Using a set square and ruler, draw lines parallel to A5B from points A4, A3, A2, and A1 to intersect line AB at points B1, B2, B3, and B4, respectively.

Result: The line segment AB is divided into five equal parts: AB4 = B4B3 = B3B2 = B2B1 = B1

B. Why it works: This construction utilizes Thales' Theorem (intercept theorem). Parallel lines intercept proportional segments on transversals.

Example: A tailor needs to divide a piece of fabric into several equal strips for a decorative border on clothing. This method ensures uniformity in the design. 2.5 Constructing a Regular Hexagon Method: Given: The length of one side, say 's'.

Procedure: Draw a line segment AB of length 's'. With A as centre and radius 's', draw a circle. With B as centre and radius 's', draw a circle. These circles intersect at O. With O as centre and radius 's' (AB), draw a circle. With A as centre and radius 's', mark off point C on the circumference of the circle.