Lesson Notes By Weeks and Term v5 - Grade 10

Lesson Video

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Performance objectives

• Bisect a line and an angle accurately.
• Construct perpendicular lines from a point on, and off, a line.
• Construct a line parallel to a given line.
• Divide a line into a specified number of equal parts.
• Apply these constructions to solve EGD problems.

Lesson summary

Engineering Graphics and Design (EGD) is a crucial subject for understanding the world around us, from the buildings we live in to the vehicles we use. Basic geometrical constructions are the foundation upon which all EGD principles are built. Think of them as the alphabet of the visual language used by engineers and designers. Without mastering these fundamentals, learners will struggle to grasp more complex concepts later on. In South Africa, where infrastructure development and manufacturing are vital for economic growth, a strong understanding of EGD and geometrical constructions is essential for contributing to these sectors.

Lesson notes

This section will explore the fundamental geometrical constructions you need to master. Remember, accuracy is key in EGD. Practice each construction repeatedly until you are confident in your ability. 2.1 Bisecting a Line Segment Bisecting a line segment means dividing it into two equal parts.

Procedure: Given line segment AB. Place the compass point at A and open the compass to a radius more than half the length of AB. Draw an arc above and below the line. Without changing the compass radius, place the compass point at B and draw arcs that intersect the arcs drawn in step

2. Label the points of intersection C and D. Draw a straight line through points C and D. The point where CD intersects AB is the midpoint of AB, and CD is the perpendicular bisector of A

B. Why it Works: The arcs create two intersecting circles with equal radii. The line connecting the intersection points of these circles is the perpendicular bisector because it passes through the midpoint of the line segment connecting the centers of the circles (A and B) and is perpendicular to it. 2.2 Bisecting an Angle Bisecting an angle means dividing it into two equal angles.

Procedure: Given angle BAC. Place the compass point at A (the vertex of the angle) and draw an arc that intersects both sides of the angle. Label the points of intersection D and E. Place the compass point at D and draw an arc inside the angle. Without changing the compass radius, place the compass point at E and draw an arc that intersects the arc drawn in step

3. Label the point of intersection

F. Draw a straight line from A through

F. This line AF bisects angle BA

C. Why it Works: The arcs create two triangles (ADF and AEF) that are congruent by SSS (side-side-side) congruence.

Therefore, angle DAF is equal to angle EAF. 2.3 Constructing a Perpendicular Line from a Point on a Line Procedure: Given line 'l' and point P on line 'l'. Place the compass point at P and draw two arcs intersecting line 'l' on either side of P. Label these intersection points A and B. Place the compass point at A and open the compass to a radius greater than AP. Draw an arc above (or below) line 'l'. Without changing the compass radius, place the compass point at B and draw an arc that intersects the arc drawn in step

3. Label the point of intersection

C. Draw a straight line from P through

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

Why it Works: This construction is based on bisecting a straight angle (180 degrees) at point P, resulting in two 90-degree angles and a perpendicular line. 2.4 Constructing a Perpendicular Line from a Point OFF a Line Procedure: Given line 'l' and point P off line 'l'. Place the compass point at P and draw an arc that intersects line 'l' at two points. Label these intersection points A and B. Place the compass point at A and open the compass to a radius greater than half the distance between A and B. Draw an arc below line 'l'. Without changing the compass radius, place the compass point at B and draw an arc that intersects the arc drawn in step

3. Label the point of intersection

C. Draw a straight line from P through

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

Why it Works: This construction essentially finds the perpendicular bisector of the line segment formed by the points where the arc from P intersects the line 'l'. 2.5 Constructing a Parallel Line to a Given Line at a Specified Distance Procedure: Given line 'l' and the desired distance 'd'. Choose two points A and B on line 'l'. At point A, construct a perpendicular line to line 'l' (using the method described in 2.3). On the perpendicular line constructed at A, measure the distance 'd' from point A and mark point C. Repeat steps 3 and 4 at point B, marking point D a distance 'd' from point B. Draw a straight line through points C and

D. This line CD is parallel to line 'l' and is a distance 'd' away.

Why it Works: This method ensures that the new line is equidistant from the original line at all points, which is the definition of parallel lines. 2.6 Dividing a Line Segment into a Specified Number of Equal Parts Procedure: Given line segment AB and the desired number of equal parts 'n'. Draw a line AC at any convenient angle from point A. Using a compass, mark 'n' equal segments along line AC, starting from point A. Label these points 1, 2, 3, ..., n. Draw a straight line from point n to point B. Using a set square and ruler, draw lines parallel to line nB from each of the points 1, 2, 3, ..., (n-1) on line A

C. These parallel lines will intersect line AB, dividing it into 'n' equal parts.

Why it Works: This construction utilizes the properties of similar triangles. The parallel lines create similar triangles, and the corresponding sides of similar triangles are proportional. Since the segments on line AC are equal, the segments on line AB will also be equal.