Lesson Notes By Weeks and Term v3 - Junior Secondary 2
Geometric Construction
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Geometric Construction
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Subject: Basic Technology
Class: Junior Secondary 2
Term: 2nd Term
Week: 1
Theme: Drawing Practice
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Define, identify,draw and listparts of a circle Use appropriateinstruments to divide a circleinto equal parts Use appropriateinstruments to draw a tangentand normal to agiven circle and two equal circles
Definition: A circle is a closed plane curve consisting of all points that are equidistant from a given fixed point called the centre.
Instruments for Drawing a Circle: A compass (also known as a pair of dividers or pair of compasses) is the primary instrument used to draw circles.
Parts of a Circle: Centre (O): The fixed point from which all points on the circumference are equidistant.
Radius (r): A straight line segment connecting the centre of the circle to any point on its circumference. All radii of a given circle are equal in length.
Diameter (d): A straight line segment that passes through the centre of the circle and has its endpoints on the circumference. The diameter is twice the length of the radius (d = 2r).
Circumference: The perimeter or distance around the circle.
Chord: Any straight line segment connecting two points on the circumference of a circle. The diameter is the longest chord.
Arc: A part of the circumference of a circle. It can be a minor arc (shorter) or a major arc (longer).
Sector: A region of a circle bounded by two radii and the arc between their endpoints. It resembles a slice of pizza. It can be a minor sector or a major sector.
Segment: A region of a circle bounded by a chord and the arc subtended by the chord. It can be a minor segment or a major segment.
Tangent: A straight line that touches the circle at exactly one point, known as the point of tangency.
Normal: A straight line that is perpendicular to the tangent at the point of tangency and passes through the centre of the circle. Essentially, it is a radius extended.
Example 1: Identifying and Labeling Parts of a Circle Teacher's
Note: Draw a large circle on the board and illustrate each part. Draw a point at the centre and label it 'O'. Draw a line from O to the circumference, label it 'r' (radius). Draw a line through O with both ends on the circumference, label it 'd' (diameter). Draw a line connecting two points on the circumference without passing through O, label it 'Chord'. Shade a region bounded by two radii and an arc, label it 'Sector'. Shade a region bounded by a chord and an arc, label it 'Segment'. Draw a line touching the circle at one point, label it 'Tangent'. Mark the point of contact 'P'. Draw a line from O through 'P' extending outwards, label it 'Normal'. Dividing a circle accurately into equal parts is fundamental in design and manufacturing. General Method for Dividing a Circle into 6 Equal Parts: Draw the circle: Using a compass, draw a circle with a desired radius and mark its centre 'O'.
Draw a diameter: Draw any diameter, say AB, passing through 'O'.
Mark arcs from endpoints: With the compass opened to the radius length, place the compass point at 'A' (one end of the diameter) and draw an arc that intersects the circumference at two points, say C and
D. Repeat from other end: Without changing the compass setting, place the compass point at 'B' (the other end of the diameter) and draw another arc that intersects the circumference at two points, say E and
F. Identify points: The six points (A, C, D, B, E, F) on the circumference divide the circle into six equal parts. These points can be joined to form a regular hexagon or used as reference for other constructions.
Dividing into 3 Equal Parts: Once divided into 6 equal parts (A, C, D, B, E, F), join alternate points. For example, join A-D, D-F, and F-A to form an equilateral triangle, effectively dividing the circumference into 3 equal parts.
Dividing into 4 Equal Parts: Draw the circle: Draw a circle with centre 'O'.
Draw a diameter: Draw any diameter, say AB, through 'O'.
Construct perpendicular diameter: Construct a perpendicular bisector to diameter AB that passes through 'O'. This will create a second diameter, say CD, perpendicular to A
B. Identify points: The four points (A, B, C, D) on the circumference divide the circle into four equal parts.
Dividing into 8 Equal Parts: Once divided into 4 equal parts (A, B, C, D), bisect the angles formed by the diameters (AOC, COB, BOD, DOA). Or, bisect each of the arcs (AC, CB, BD, DA) using a compass. This will yield 8 equal points. Dividing into other Equal Parts (e.g., 5, 7, 10): For divisions not easily achieved with basic compass constructions (e.g., 5, 7, 10), a protractor is typically used.
Method: Divide 360 degrees by the desired number of parts (e.g., for 5 parts: 360/5 = 72 degrees). Use the protractor to mark off these angles from the centre of the circle along the circumference. Teacher's
Note: Emphasize the importance of accuracy and using a sharp pencil.
Definition of Tangent: A tangent is a line that touches a circle at only one point, called the point of tangency.
Property: A tangent to a circle is always perpendicular to the radius drawn to the point of tangency.
Construction 1: Drawing a Tangent to a Circle at a Given Point on its Circumference Draw the circle: Draw a circle with centre 'O' and mark a point 'P' on its circumference where the tangent is to be drawn.
Draw the radius: Draw a radius from 'O' to 'P'.
Extend the radius: Extend the line OP beyond P to a point Q. (This forms a line segment OPQ).
Construct perpendicular at P: Place the compass point at P and draw arcs of equal radius on both sides of P along the line OPQ (e.g., mark points X and Y on the line OPQ).
Bisect the segment: With the compass opened to a radius greater than PX (or PY), place the compass point at X and draw an arc above/below P. Repeat with the compass point at Y, drawing another arc to intersect the first one. Let the intersection point be
R. Draw the tangent: Draw a straight line through P and
R. This line PR is the tangent to the circle at point
P. Construction 2: Drawing a Normal to a Circle at a Given Point on its Circumference Follow steps 1 and 2 from Construction 1: Draw the circle with centre 'O' and mark point 'P' on its circumference. Draw the radius O
P. Draw the normal: The line segment OP, when extended through P, is the normal to the circle at point
P. The normal is simply the radius extended outwards from the point of tangency.
Construction 3: Drawing Common External Tangents to Two Equal Circles Draw the circles: Draw two equal circles of given radius (e.g., 3cm) with centres O1 and O2, separated by a given distance (e.g., 8cm). Draw the line connecting O1 and O
2. Draw perpendicular radii: At O1, construct a line perpendicular to O1O2, intersecting the circumference at A and B. Similarly, at O2, construct a line perpendicular to O1O2, intersecting the circumference at C and D. (Ensure A and C are on the same "side" of the O1O2 line, and B and D are on the other "side").
Draw the tangents: Join A to C with a straight line. AC is one common external tangent. Join B to D with a straight line. BD is the other common external tangent.
Reasoning: Since the circles are equal, the radii perpendicular to the line joining the centres will be parallel and equal. Connecting their endpoints creates parallel tangents. This section provides in-depth explanations of the core concepts, definitions, and construction methods required for the topic.
Geometric construction is integral to many aspects of Nigerian life and industry: Architecture and Urban Planning (e.g., Housing, Roads): Application: Architects and civil engineers use geometric construction to design circular elements in buildings such as arches, domes (common in mosques and modern structures), circular windows, and spiral staircases. In urban planning, the design of roundabouts for traffic management relies heavily on accurate circle construction and tangent concepts for smooth traffic flow.
Nigerian Context: The construction of new estates often features circular landscapes or driveways. The layout of market stalls or community gathering points might incorporate circular arrangements, requiring precise division of space. Roads in cities like Abuja or Lagos frequently use curves and tangents, which are geometrically constructed. Art, Craft, and Traditional Designs (e.g., Adire, Pottery, Calabash Carving): Application: Many traditional Nigerian crafts feature intricate patterns that originate from geometric principles. Artisans in Adire fabric dyeing, calabash carving, or pottery making use concepts of dividing circles into equal segments to create symmetrical and aesthetically pleasing designs. The making of traditional beads or jewelry also involves precise circular shapes.
Nigerian Context: A calabash carver in Oyo state might mentally or physically divide a circular calabash into sections to ensure a balanced design before carving. A weaver producing circular mats or baskets in the Niger Delta area needs to ensure equal divisions for structural integrity and pattern consistency. Mechanical and Technical Engineering (e.g., Workshops, Industries): Application: In workshops and industries across Nigeria, geometric construction is vital for the design and fabrication of mechanical components. Gears, pulleys, bearings, and shafts are circular objects whose precise dimensions, division into teeth (for gears), and positioning rely on these principles. Drawing tangents is essential for designing belt drives and determining clearance for moving parts.
Nigerian Context: A mechanic in a local automobile repair shop in Kaduna needs to understand circular geometry when replacing a wheel bearing or working with engine gears. Artisans in Aba producing shoes or leather goods may use circular templates that require geometric construction for consistent product quality.