Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Construct a regular pentagon given the length of one side.
• Divide a line segment into a given number of equal parts.
• Construct a tangent to a circle from a point outside the circle.
• Apply these constructions to solve practical drawing problems.

Lesson summary

This week, we delve deeper into advanced geometrical constructions, building upon the foundational skills learned in Grade 10 and Week 1 of Grade

1

1. Mastery of these constructions is crucial for success in Engineering Graphics and Design as they form the basis for accurately representing complex shapes and objects in various engineering disciplines. In South Africa, these skills are particularly relevant in industries like construction, manufacturing, and design, where precise and accurate drawings are essential for effective communication and project execution.

Lesson notes

2.1 Constructing a Regular Pentagon Given the Length of One Side A regular pentagon has five equal sides and five equal angles. Constructing it accurately requires understanding angle bisection and circle construction.

Method 1: Using a Circle Step 1: Draw the Given Side: Begin by drawing the given side A

B. Let's say AB = 50mm.

Step 2: Extend the line AB: Extend the line AB on both sides.

Step 3: Construct a Perpendicular: At point A, construct a line perpendicular to A

B. Step 4: Locate the midpoint of AB: Find the midpoint of AB by using a compass and bisecting the line. Mark this midpoint as

M. Step 5: Mark a point on the perpendicular line: Using M as the centre, set your compass radius to AM. Draw an arc intersecting the perpendicular line you drew in step

3. Mark this intersection point as

N. This now forms a right angled triangle AM

N. Step 6: Set your compass to length AB: Using point A as the centre, set your compass to the length of A

B. Draw an arc intersecting line AB extended. Label the point

P. Step 7: Draw a line from N through P and extend it.

Step 8: Using compass draw a circle with centre A and radius equal to A

B. This is the radius length of the pentagon's side.

Step 9: Find intersection point C: With centre N and radius = AB, draw an arc intersecting the extended line drawn in step

7. Mark the point of intersection as

C. AC is another side of the pentagon Step 10: Find point E: Set your compass length to AB. With centre B, draw an arc. With centre C draw an arc intersecting the previous arc. Label the intersection point as

E. Step 11: Find point D: Set your compass length to AB. With centre C draw an arc. With centre A draw an arc intersecting the previous arc. Label the intersection point as

D. Step 12: Join the vertices: Join points A, D, E, C and B to form the regular pentagon.

Why this works: This method leverages geometric properties and the relationship between angles and sides in a regular pentagon. The perpendicular and bisection help to create accurate angles needed for construction.

Example: Let's say we need to construct a regular pentagon with a side length of 45mm using the method described above. Follow all the steps and ensure accuracy while measuring and drawing. 2.2 Dividing a Line Segment into Equal Parts Dividing a line segment into equal parts is a fundamental skill.

There are two primary methods: Geometric division and Mathematical calculation combined with accurate measurement.

Method 1: Geometric Division (Parallel Line Method)

Step 1: Draw the Line Segment: Draw the line segment AB that you want to divide. For example, let AB = 100mm.

Step 2: Draw an Inclined Line: Draw a line AC at any convenient angle from point

A. Step 3: Mark Equal Divisions: On line AC, mark the number of equal divisions required (e.g., 5) using a compass. Ensure each division is of equal length. Let's mark these points as 1, 2, 3, 4, and

5. The length of each division is arbitrary; consistency is key.

Step 4: Connect the Last Division: Connect the last division (5) on line AC to point B on line A

B. Step 5: Draw Parallel Lines: Using a set square and T-square (or parallel rule), draw lines parallel to line 5B from each of the division points (4, 3, 2, 1) on line A

C. These parallel lines will intersect line A

B. Step 6: Mark the Divisions: The points where the parallel lines intersect line AB will divide AB into the required number of equal parts.

Method 2: Calculation and Measurement Step 1: Measure the Length: Measure the total length of the line segment AB accurately. For example, AB = 125mm.

Step 2: Calculate the Division Length: Divide the total length by the number of equal parts required. If we need to divide it into 5 equal parts, the length of each part is 125mm / 5 = 25mm.

Step 3: Mark the Divisions: Using a ruler, measure and mark points along line AB at intervals of 25mm.

Why this works: The geometric division method is based on the principle that parallel lines cut off proportional segments on transversals. The calculation method is simply applying division to the total length.

Example: Let's divide a line segment of 75mm into 3 equal parts using both geometric division and measurement.

Geometric Division: Follow the steps above. Accuracy with the parallel lines is vital.

Calculation: 75mm / 3 = 25mm. Measure and mark at 25mm and 50mm. 2.3 Constructing a Tangent to a Circle from a Point Outside the Circle A tangent to a circle is a line that touches the circle at only one point.

Step 1: Draw the Circle and the Point: Draw a circle with centre O and mark a point P outside the circle.

Step 2: Join the Centre to the Point: Draw a line segment connecting the centre O of the circle to the external point

P. Step 3: Bisect the Line: Bisect the line segment O

P. Find the midpoint M of OP using a compass.

Step 4: Draw a New Circle: Using M as the centre and OM (or MP) as the radius, draw a new circle.