Lesson Notes By Weeks and Term v3 - Junior Secondary 3
Similar Shapes
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Similar Shapes
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Subject: General Mathematics
Class: Junior Secondary 3
Term: 3rd Term
Week: 6
Theme: Mensuration And Geometry
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Identify similar figures:- triangles rectangles, squares cubes and cuboids Identify the presence of similar shapes in the environment Enlarge figures using scale factors Calculate lengths, are as and volumes of similar figures Solve problems on quantitative reasoning in volving similar shapes.
Two triangles are similar if any of the following conditions are met: Angle-Angle-Angle (AAA)
Similarity: If all three corresponding angles are equal. (In practice, AA is sufficient, as the third angle will automatically be equal).
Side-Side-Side (SSS)
Similarity: If the ratios of all three corresponding sides are equal.
Side-Angle-Side (SAS)
Similarity: If two corresponding sides are proportional, and the included angles are equal.
Properties of Similar Triangles: Given two similar triangles, $\triangle ABC \sim \triangle DEF$: $\angle A = \angle D$, $\angle B = \angle E$, $\angle C = \angle F$ $\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k$ (where k is the scale factor)
Example 1: Identifying Similar Triangles Problem: Determine if the triangles below are similar.
Triangle PQR: P(90°), Q(60°), R(30°)
Triangle XYZ: X(90°), Y(60°), Z(30°)
Explanation: Since all corresponding angles are equal ($\angle P = \angle X = 90^\circ$, $\angle Q = \angle Y = 60^\circ$, $\angle R = \angle Z = 30^\circ$), the triangles $\triangle PQR$ and $\triangle XYZ$ are similar by AAA similarity criterion.
Example 2: Finding Unknown Side Lengths in Similar Triangles Problem: Two triangles, $\triangle ABC$ and $\triangle DEF$, are similar. $AB = 6$ cm, $BC = 8$ cm, $AC = 10$ cm. If $DE = 9$ cm, find the lengths of $EF$ and $DF$.
Solution: Since $\triangle ABC \sim \triangle DEF$, the ratio of corresponding sides is constant. $\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$ Calculate the scale factor (k) using the known corresponding sides $AB$ and $DE$: $k = \frac{DE}{AB} = \frac{9}{6} = \frac{3}{2}$ or $1.5$ Find $EF$: $\frac{BC}{EF} = \frac{1}{k} \Rightarrow \frac{8}{EF} = \frac{1}{1.5} \Rightarrow EF = 8 \times 1.5 = 12$ cm. Alternatively, $EF = BC \times k = 8 \times \frac{3}{2} = 12$ cm. Find $DF$: $\frac{AC}{DF} = \frac{1}{k} \Rightarrow \frac{10}{DF} = \frac{1}{1.5} \Rightarrow DF = 10 \times 1.5 = 15$ cm. Alternatively, $DF = AC \times k = 10 \times \frac{3}{2} = 15$ cm.
Conclusion: $EF = 12$ cm and $DF = 15$ cm. For any two polygons to be similar, two conditions must be met: Corresponding angles are equal. The ratio of corresponding sides is constant (i.e., the scale factor is the same for all pairs of corresponding sides).
Squares: All squares are similar to each other because all angles are 90 degrees, and the ratio of sides is always constant (1:1 for squares, but any two squares are similar, just their side length ratio will differ).
Rectangles: Two rectangles are similar if the ratio of their corresponding side lengths (length to length, width to width) is constant.
Example 3: Similar Rectangles Problem: A rectangular room in a Nigerian house has dimensions 4 meters by 6 meters. A scaled-down drawing of the room has dimensions 10 cm by 15 cm. Are the room and its drawing similar? If so, what is the scale factor?
Solution: Convert units to be consistent. Let's use cm.
Room dimensions: 4 m = 400 cm, 6 m = 600 cm.
Drawing dimensions: 10 cm, 15 cm. Check the ratio of corresponding sides (length to length, width to width).
Ratio of lengths: $\frac{600 \text{ cm}}{15 \text{ cm}} = 40$ Ratio of widths: $\frac{400 \text{ cm}}{10 \text{ cm}} = 40$ Since the ratios are equal, the room and its drawing are similar. The scale factor from the drawing to the actual room is $k = 40$. (Or from room to drawing: $1/40$). Two solids are similar if they have the same shape and their corresponding linear dimensions are in proportion.
This means: The ratio of corresponding lengths (length, width, height, radius, etc.) is constant (the scale factor, k).
Cubes: All cubes are similar to each other.
Cuboids: Two cuboids are similar if the ratio of their corresponding lengths, widths, and heights are all equal.
Example 4: Similar Cuboids Problem: A large storage cuboid has dimensions 30 cm x 20 cm x 10 cm. A smaller, similar cuboid is to be made with a length of 12 cm. Find the width and height of the smaller cuboid.
Solution: The dimensions of the large cuboid are $L_1 = 30$ cm, $W_1 = 20$ cm, $H_1 = 10$ cm. The length of the smaller cuboid is $L_2 = 12$ cm. Calculate the scale factor (k) from the large to the small cuboid: $k = \frac{L_2}{L_1} = \frac{12}{30} = \frac{2}{5} = 0.4$ Find the width of the smaller cuboid ($W_2$): $W_2 = W_1 \times k = 20 \text{ cm} \times 0.4 = 8$ cm. Find the height of the smaller cuboid ($H_2$): $H_2 = H_1 \times k = 10 \text{ cm} \times 0.4 = 4$ cm.
Conclusion: The smaller cuboid has dimensions 12 cm x 8 cm x 4 cm. The scale factor (k) is the ratio of a length on the new figure to the corresponding length on the original figure. If $k > 1$, the figure is enlarged. If $0 < k < 1$, the figure is reduced. If $k = 1$, the figures are congruent (same size and shape).
Architecture and Construction: Architects and engineers use similarity extensively when designing buildings. They create scaled-down models or blueprints (floor plans) which are similar to the actual structures. Understanding scale factors allows them to calculate actual dimensions, material quantities (area for roofing/flooring, volume for concrete), and costs from these scaled drawings. For instance, in constructing a new market in Lagos, understanding similarity helps estimate materials for various stalls based on a master plan.
Cartography (Map Making) and Geography: Maps are scaled-down representations of actual geographical areas (states, countries, continents). Every map has a scale factor (e.g., 1:50,000 or 1 cm = 10 km) that indicates the ratio of a distance on the map to the corresponding actual distance on the ground. This knowledge is essential for navigation, land surveying, urban planning, and interpreting satellite images of Nigerian landscapes (e.g., assessing land use patterns in rural areas or urban expansion in Abuja).
Photography and Digital Imaging: When an image is enlarged or reduced (e.g., from a mobile phone picture to a print-out poster of a Nigerian cultural event), the original and modified images are similar. The aspect ratio (ratio of width to height) must be maintained to avoid distortion. Understanding scale factors helps photographers crop images correctly or predict the quality of enlarged prints. This also applies to designing logos or patterns for Nigerian fabric (Ankara, Adire) where a small design is replicated or scaled.