Similarity and Congruence
Similarity and congruence are key geometric concepts for comparing shapes. This guide will explore what makes shapes identical (congruent) or just scaled versions of each other (similar), and reveal the crucial link between their lengths, areas, and volumes.
Congruence vs. Similarity
Congruent Shapes ($\cong$)
Two shapes are congruent if they are exactly the same shape and size. Think of them as perfect duplicates. All corresponding sides and angles are equal.
Similar Shapes ($\sim$)
Two shapes are similar if they have the same shape but can be different sizes. One is an enlargement of the other. Corresponding angles are equal, and corresponding sides are in the same ratio.
The Key Relationship: Length, Area, and Volume
When you scale a shape, its lengths, areas, and volumes change in a predictable way. If the linear scale factor is $k$, then:
| Measurement | Scale Factor | Ratio |
|---|---|---|
| Length | $k$ | $a : b$ |
| Area | $k^2$ | $a^2 : b^2$ |
| Volume | $k^3$ | $a^3 : b^3$ |
Worked Examples
Example 1: Finding a Missing Length
Two similar rectangles are shown. Find the length $x$.
- Find the linear scale factor ($k$): Use the known corresponding heights. $k = \frac{6}{4} = 1.5$.
- Calculate the missing length: $x = 8 \times 1.5 = 12$ cm.
Answer: $x = 12$ cm
Example 2: Finding an Area
Two similar triangles have corresponding sides of 5 cm and 10 cm. If the area of the smaller triangle is 12 cm², what is the area of the larger one?
- Find the linear scale factor ($k$): $k = \frac{10}{5} = 2$.
- Find the area scale factor: Area factor $= k^2 = 2^2 = 4$.
- Calculate the new area: Area $= 12 \text{ cm}^2 \times 4 = 48 \text{ cm}^2$.
Answer: 48 cm²
Example 3: Finding a Length from Volumes
Two similar cones have volumes of 8 cm³ and 27 cm³. If the height of the smaller cone is 4 cm, what is the height of the larger cone?
- Find the volume scale factor: $\frac{27}{8}$.
- Find the linear scale factor ($k$): $k = \sqrt[3]{\frac{27}{8}} = \frac{3}{2} = 1.5$.
- Calculate the new height: Height $= 4 \text{ cm} \times 1.5 = 6$ cm.
Answer: 6 cm
Tutor Insights
🤔 Common Misunderstandings
- Forgetting to square or cube the scale factor. The most common mistake — students find $k$ and apply it directly to areas or volumes.
- Using the wrong scale factor direction. When working backwards from area or volume to find a length, you must take the square root or cube root of the scale factor.
📝 Common Exam Mistakes
- Mixing up “New” and “Old” lengths when calculating the scale factor.
- Simple arithmetic errors with squaring, cubing, or finding roots.
- Not showing working, especially how you found the linear, area, and volume scale factors.
Practice Questions
- Two similar cuboids have corresponding lengths in the ratio $1:3$. The surface area of the smaller cuboid is 20 cm². What is the surface area of the larger cuboid?
- Two similar vases have heights in the ratio $2:5$. The larger vase has a volume of 3750 cm³. Calculate the volume of the smaller vase.
Show Answers
- Working: Length ratio $1:3$. Area ratio $1^2:3^2 = 1:9$. Area of larger $= 20 \text{ cm}^2 \times 9$.
Answer: 180 cm². - Working: Length ratio $2:5$. Volume ratio $2^3:5^3 = 8:125$. $\frac{V_S}{3750} = \frac{8}{125} \implies V_S = \frac{8}{125} \times 3750$.
Answer: 240 cm³.
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