Lesson Notes By Weeks and Term v3 - Senior Secondary 1
Enlargement and Reduction of Figures
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Enlargement and Reduction of Figures
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Subject: Technical Drawings
Class: Senior Secondary 1
Term: 3rd Term
Week: 10
Theme: Geometrical Constructions
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Explain the applications of enlargement and reduction of objects Reduce given figures to given proportions/ratios. Enlarge given figures to given ratios.
This method involves selecting a point (the centre of enlargement/reduction) and drawing lines from this point through each vertex of the original figure. The corresponding vertices of the scaled figure lie along these lines at a proportional distance from the centre. Steps for Enlargement (e.g., Ratio 2:1)
Choose a Centre Point (O): This point can be inside, outside, or on one of the vertices of the original figure. Its position affects the location of the enlarged figure but not its size or shape.
Draw the Original Figure: Accurately draw the given figure (e.g., triangle ABC).
Draw Radial Lines: From the chosen centre point O, draw straight lines passing through each vertex of the original figure (e.g., O-A, O-B, O-C). Extend these lines sufficiently.
Measure and Mark New Vertices: Measure the distance from O to each vertex of the original figure (OA, OB, OC).
To enlarge by a ratio of 2:1, multiply these distances by
2. Mark A' on line O-A such that OA' = 2 OA. Mark B' on line O-B such that OB' = 2 O
B. Mark C' on line O-C such that OC' = 2 O
C. Connect New Vertices: Connect the new marked points (A', B', C') to form the enlarged figure. The new figure A'B'C' will be similar to ABC but twice as large.
Example 1 (Enlargement): Enlarge Triangle ABC with vertices A(1,1), B(3,1), C(2,3) by a ratio of 2:1, using the origin (0,0) as the centre of enlargement.
Step 1: Plot A(1,1), B(3,1), C(2,3) and draw triangle AB
C. Choose O(0,0) as the centre.
Step 2: Draw lines from O through A, B, and
C. Step 3: OA = sqrt((1-0)^2 + (1-0)^2) = sqrt(2). OA' = 2 OA = 2 * sqrt(2). OB = sqrt((3-0)^2 + (1-0)^2) = sqrt(10). OB' = 2 OB = 2 * sqrt(10). OC = sqrt((2-0)^2 + (3-0)^2) = sqrt(13). OC' = 2 OC = 2 * sqrt(13). Alternatively, simply multiply the coordinates by the ratio: A' = (12, 1*2) = (2,2) B' = (32, 1*2) = (6,2) C' = (22, 3*2) = (4,6)
Step 4: Connect A'(2,2), B'(6,2), C'(4,6) to form the enlarged triangle A'B'C'. Steps for Reduction (e.g., Ratio 1:2 or 3:4)
Choose a Centre Point (O): As with enlargement, this point can be anywhere.
Draw the Original Figure: Accurately draw the given figure.
Draw Radial Lines: From the chosen centre point O, draw straight lines passing through each vertex of the original figure (e.g., O-A, O-B, O-C).
Measure and Mark New Vertices: Measure the distance from O to each vertex of the original figure (OA, OB, OC).
To reduce by a ratio of 1:2, divide these distances by
2. For a ratio of 3:4, multiply by 3/
4. Mark A' on line O-A such that OA' = (1/2)
OA (for 1:2 ratio). Mark B' on line O-B such that OB' = (1/2) O
B. Mark C' on line O-C such that OC' = (1/2) O
C. Connect New Vertices: Connect the new marked points (A', B', C') to form the reduced figure. The new figure A'B'C' will be similar to ABC but half its size.
Example 2 (Reduction): Reduce a rectangular plot ABCD with A(6,4), B(10,4), C(10,8), D(6,8) by a ratio of 1:2, using the origin (0,0) as the centre of reduction.
Step 1: Plot A(6,4), B(10,4), C(10,8), D(6,8) and draw rectangle ABC
D. Choose O(0,0) as the centre.
Step 2: Draw lines from O through A, B, C, and
D. Step 3: Multiply the coordinates by the ratio (1/2): A' = (60.5, 4*0.5) = (3,2) B' = (100.5, 4*0.5) = (5,2) C' = (100.5, 8*0.5) = (5,4) D' = (60.5, 8*0.5) = (3,4)
Step 4: Connect A'(3,2), B'(5,2), C'(5,4), D'(3,4) to form the reduced rectangle A'B'C'D'. --- This method is particularly useful for irregular shapes or figures with curves, as it allows for the transfer of features square by square. Steps for Enlargement (e.g., Ratio 3:2)
Draw Original Figure and Grid: Draw the original figure (e.g., a logo or a drawing of a local craft) on a sheet of paper. Overlay it with a grid of equally sized squares (e.g., 10mm x 10mm). Number or label the rows and columns.
Draw New Grid: On another sheet or adjacent to the original, draw a new grid.
If enlarging by 3:2, the side length of each square in the new grid should be 3/2 times (1.5 times) the side length of the squares in the original grid (e.g., 15mm x 15mm squares). Maintain the same number of rows and columns. Label the new grid identically.
Transfer Points/Features: Carefully observe how the original figure intersects or fills each square in the original grid. Transfer these points and segments to the corresponding squares in the new, larger grid. For curved lines, focus on key points along the curve and their positions within each square.
Connect Points: Connect the transferred points to form the enlarged figure.
Example 3 (Enlargement): Enlarge an irregular quadrilateral ABCD by a ratio of 3:2 using the grid method.
Step 1: Draw the irregular quadrilateral ABC
D. Overlay it with a grid of 10mm x 10mm squares.
Step 2: Draw a new grid next to it, where each square is 15mm x 15mm (10mm 3/2). Ensure the number of squares is the same.
Step 3: For each square in the original grid, identify the part of the quadrilateral that falls within it. For example, if line AB passes through the middle of square (1,2) and the top-left corner of square (2,2), mark corresponding points in square (1,2)' and (2,2)' of the new grid.
Step 4: Connect these marked points in the new grid to form the enlarged irregular quadrilateral A'B'C'D'. Steps for Reduction (e.g., Ratio 1:2)
Draw Original Figure and Grid: Draw the original figure and overlay it with a grid of equally sized squares.
Draw New Grid: Draw a new grid where each square's side length is proportionally smaller (e.g., for 1:2 reduction, if original squares are 10mm, new squares are 5mm x 5mm). Maintain the same number of rows and columns.
Transfer Points/Features: Transfer the features of the original figure from its grid to the corresponding, smaller squares in the new grid.
Connect Points: Connect the transferred points to form the reduced figure.
Example 4 (Reduction): Reduce a complex architectural detail of a traditional Nigerian window by a ratio of 1:2 using the grid method.
Step 1: Draw the architectural detail. Overlay it with a grid of 20mm x 20mm squares.
Step 2: Draw a new grid, where each square is 10mm x 10mm (20mm 1/2).
Step 3: Carefully transfer the intricate lines and curves of the window detail, square by square, from the larger grid to the smaller grid.
Step 4: Connect the transferred points to form the reduced architectural detail. --- This method is primarily used for scaling a single line or for finding proportional lengths. It can be extended to scale figures by scaling individual dimensions or features. Steps for Scaling a Line Segment (e.g., Reduce line AB by 3:4)
Draw Original Line: Draw the line segment AB to be scaled.
Draw a Construction Line: From one end of AB (e.g., A), draw an auxiliary line at any convenient acute angle.
Mark Proportions: Along the auxiliary line, mark off a number of equal divisions corresponding to the denominator of the ratio (e.g., 4 divisions for 3:4 ratio). Label them 1, 2, 3,
4. Connect to End: Connect the last division point (e.g., 4) to the other end of the original line (e.g., B).
Draw Parallel Lines: From the division point corresponding to the numerator of the ratio (e.g., 3), draw a line parallel to line 4B, intersecting the original line AB at a new point (e.g., B'). The segment AB' will be the scaled length (3/4 of AB). This method can be applied to scale each side of a polygon, and then construct the new polygon using these scaled side lengths.
However, it's generally more complex for scaling entire figures compared to the radial or grid methods, unless the figure is simple and easily defined by its linear dimensions. Introduction to Scaling Scaling in technical drawing refers to the process of representing an object at a size different from its actual size while maintaining its true proportions. This involves either enlargement (making the drawing larger than the actual object) or reduction (making the drawing smaller than the actual object). The relationship between the drawing size and the actual object size is expressed as a ratio or proportion.
Enlargement: When a figure is drawn larger than its original size.
The scale ratio is typically written as X:1, where X is greater than 1 (e.g., 2:1, 5:1, meaning the drawing is 2 or 5 times larger than the actual object).
Reduction: When a figure is drawn smaller than its original size.
The scale ratio is typically written as 1:X, where X is greater than 1 (e.g., 1:2, 1:100, meaning the drawing is half or one-hundredth the size of the actual object).
Proportion/Ratio: A comparison of two quantities. In scaling, it defines how much larger or smaller the new figure will be compared to the original. For example, a ratio of 2:1 means the new figure's dimensions are twice those of the original.
A ratio of 1:2 means the new figure's dimensions are half those of the original.
A ratio of 3:4 means the new figure's dimensions are three-quarters of the original (a reduction). Methods of Enlargement and Reduction There are several methods for enlarging and reducing figures, each suitable for different types of figures and contexts.
The most common methods are: Radial Method (Centre of Enlargement/Reduction) Grid Method (Squaring Method) Parallel Lines Method (Proportional Division) ---
Architecture and Construction (Nigerian Housing & Infrastructure): Application: Architects and civil engineers in Nigeria frequently use enlargement and reduction. For instance, a detailed plan of a proposed residential building in Abuja might be drawn at a scale of 1:100 (reduction) for overall layout and presentation.
However, specific intricate details like window frames, door designs, or plumbing layouts for a typical 'face-me-I-face-you' apartment block or a modern duplex would require partial enlargement (e.g., 2:1 or 5:1) to show precise dimensions and construction specifics to local craftsmen and builders. This is crucial for accurate construction of essential public infrastructure like bridges, hospitals, and schools across the country. Cartography and Urban Planning (Mapping Nigerian Landscapes): Application: Geographers and urban planners use scaling extensively when creating maps of Nigerian states, cities (e.g., Lagos, Kano, Port Harcourt), or local government areas. A national road map would be a massive reduction (e.g., 1:1,000,000), showing major highways and cities. For detailed urban planning within a specific local government area (LGA), say in Ibadan, a section of the map might be enlarged to 1:5,000 to show individual plots, streets, and proposed developments like markets or public parks more clearly. Manufacturing and Fabrication (Local Industries): Application: In Nigeria's growing manufacturing sector and local fabrication workshops (e.g., vehicle parts, agricultural tools, furniture), scaling is vital. A designer might create an initial concept drawing of a new component for a locally assembled vehicle or a machine for processing agricultural products (like garri processing machines). This initial drawing might be at a convenient scale (e.g., 1:5 reduction). When sending the design to the workshop, a critical part that requires high precision (e.g., a gear or a bracket) could be enlarged to a 2:1 or even 5:1 scale to guide the skilled fabricators and machinists in their work, ensuring accuracy and functionality.