Lesson Notes By Weeks and Term v5 - Grade 11
Advanced geometrical constructions – Week 1 focus
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Advanced geometrical constructions – Week 1 focus
Download the Lessonotes Mobile South Africa app for faster lesson access on Android and iPhone.
Subject: Engineering Graphics and Design
Class: Grade 11
Term: 1st Term
Week: 1
Theme: General lesson support
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Engineering Graphics and Design (EGD) is a visual language vital in South Africa's engineering, manufacturing, and construction sectors. Mastery of advanced geometrical constructions is not just about drawing lines; it's about developing spatial reasoning, problem-solving abilities, and the precision needed to translate ideas into tangible realities. This week, we'll focus on the fundamental advanced geometrical constructions that underpin more complex EGD applications. Imagine designing affordable housing solutions in a township, planning the layout of a factory floor in Gauteng, or creating components for renewable energy infrastructure – these all rely on the principles we'll explore.
This week, we'll delve into specific geometrical constructions.
Let's break them down: 2.
1. Regular Polygons within and circumscribed about a circle: Concept: Regular polygons are polygons with all sides and angles equal. Constructing them accurately relies on dividing a circle into equal arcs.
Pentagon: Within the circle:* Accurately bisect a radius of the circle. Using the midpoint of the radius as a center, swing an arc from the intersection of the radius and the circle to the diameter perpendicular to the first radius. The distance from where that arc intersects the diameter to the intersection of the first radius and the circle is the length of one side of the pentagon.
Circumscribed about the circle:* Draw tangents to the circle at points equally spaced around its circumference (72 degrees apart, which can be found by 360/5). These tangents will intersect to form the pentagon.
Hexagon: Within the circle:* The radius of the circle is equal to the length of one side of the hexagon. Step off the radius distance around the circumference of the circle six times to find the vertices of the hexagon.
Circumscribed about the circle:* Draw tangents to the circle at points equally spaced around its circumference (60 degrees apart, which can be found by 360/6).
Octagon: Within the circle:* Bisect the angles formed by two perpendicular diameters. This creates angles of 45 degrees. Connect the points where these lines intersect the circle to form the octagon.
Circumscribed about the circle:* Draw tangents to the circle at points equally spaced around its circumference (45 degrees apart, which can be found by 360/8). 2.
2. Dividing a Straight Line into Equal Parts: Concept: This doesn't involve measuring; it's about creating proportional segments.
Method: Draw a line at any convenient angle from one end of the given line. Using a compass, mark off the desired number of equal spaces along this angled line. (e.g., if you need to divide into 5 parts, mark off 5 equal spaces). Connect the last point on the angled line to the other end of the original line. Draw lines parallel to this connecting line from each of the other marked points on the angled line. These parallel lines will divide the original line into the desired number of equal parts.
Why it works: This construction uses similar triangles to create proportional division. 2.
3. Constructing a Tangent to a Circle from a Point Outside the Circle: Concept: A tangent touches a circle at only one point and is perpendicular to the radius at that point.
Method: Join the given external point to the center of the circle. Bisect this line (find its midpoint). Using the midpoint as the center, draw a circle with a radius equal to half the length of the line joining the external point and the center of the first circle. The points where this new circle intersects the original circle are the points of tangency. Draw lines from the external point to these points of tangency. These are the tangents.
Why it works: The angle inscribed in a semicircle is always a right angle. This construction ensures that the angle between the radius and the tangent line is 90 degrees. 2.
4. Constructing Arcs Tangent to Two Lines: Concept: The center of the tangent arc must be equidistant from both lines.
External Tangency: Draw lines parallel to the given lines, at a distance equal to the radius of the desired tangent arc. The intersection of these parallel lines is the center of the tangent arc. Draw the arc with the specified radius.
Internal Tangency: Determine if the given lines intersect. If so, find their intersection point. If they are parallel, the technique can still be used with slightly different steps. Bisect the angle (if any) created between the two lines. Determine the radius of the tangent arc. Measure from the intersection of the two lines, along one of the lines, a distance equal to x. Then, from that point, draw a perpendicular line to that line. Measure from the first line, along the perpendicular line you just drew, the length of the radius. The end of that length is the center point of your tangent arc. Draw an arc with the specified radius between the two given lines, with the arc beginning and ending exactly where the arc is tangent to the two lines.