Study Notes: Geometry - Similarity (0580 IGCSE Mathematics)
Hey there! Welcome to the chapter on Similarity. Don't worry, this topic is easier than it sounds. If you can understand how a photocopier or a map works, you already understand similarity!
Similarity in Geometry is all about two shapes having the exact same shape, but possibly different sizes. They are perfect scaled copies of each other. This is extremely useful for calculating distances and volumes that are impossible to measure directly, like the height of a skyscraper!
1. Defining Similar Shapes
What Does 'Similar' Mean?
Two shapes are mathematically similar if they satisfy two key conditions:
- Corresponding angles are equal.
- Corresponding sides are in proportion (meaning they have the same ratio).
Analogy: Think about taking a digital photograph and zooming in. The zoomed image is similar to the original. Every angle is the same, but every side length has been multiplied by the same scaling number.
Quick Contrast: Similarity vs. Congruence
In the syllabus, you need to understand both terms:
- Congruent Shapes: Exactly the same shape and the same size. (A scale factor of 1).
- Similar Shapes: Exactly the same shape, but possibly a different size.
Key Takeaway: If two shapes are similar, their angles are identical, and their side lengths are related by a single, constant multiplier called the Scale Factor.
2. The Length Scale Factor (LSF), \(k\)
The core concept of similarity is the Length Scale Factor (LSF), usually denoted by \(k\).
Calculating the Length Scale Factor (k)
The scale factor \(k\) is the ratio of corresponding lengths between the two shapes.
\[ k = \frac{\text{New Length (or Larger Length)}}{\text{Original Length (or Smaller Length)}} \]
Important Tip: Always be clear about which shape is the 'New' one and which is the 'Original' one. If you go from a small shape to a large shape, \(k > 1\). If you go from a large shape to a small shape, \(k < 1\) (it will be a fraction or decimal).
Step-by-Step: Finding an Unknown Length
If you know two shapes (A and B) are similar, you can find any unknown side length if you can find the scale factor.
- Identify Corresponding Sides: Find two sides, one on Shape A and one on Shape B, that correspond (meaning they are in the same position on both shapes) AND whose lengths are both known.
- Calculate the Scale Factor (k): Use the formula above to find \(k\).
- Calculate the Unknown Length:
\[ \text{Unknown Length} = \text{Corresponding Known Length} \times k \]
Example: Triangle P has a side of 5 cm. Similar Triangle Q has a corresponding side of 20 cm. If Triangle P has another side of 8 cm, what is the matching side on Q?
- Step 1 & 2 (Find k): Going from P to Q (small to large). \( k = \frac{20}{5} = 4 \).
- Step 3 (Calculate unknown): Unknown side \( = 8 \times 4 = 32 \) cm.
Key Takeaway: All lengths (sides, perimeters, heights, diameters) in similar shapes use the Length Scale Factor (\(k\)).
3. Similarity and Area (Extended Content - E5.3)
This is where students often make mistakes! When a shape is enlarged by a length scale factor \(k\), the area is NOT simply multiplied by \(k\).
If the Length Scale Factor is \(k\), the Area Scale Factor is \(k^2\).
The Area Relationship:
\[ \frac{\text{Area of New Shape}}{\text{Area of Original Shape}} = k^2 \]
Analogy: Imagine a square with side 3 cm. Area = 9 cm². If we use a length scale factor \(k=2\) (sides become 6 cm), the new area is 36 cm². Notice that \(9 \times 2 = 18\), but \(9 \times 4 = 36\). The area increased by \(k^2 = 2^2 = 4\).
Using the Area Scale Factor
To find an unknown area, you must first find the LSF \(k\) using lengths, then square it to get \(k^2\).
Step-by-Step for Area:
- Find the LSF (\(k\)): Use two corresponding known lengths (e.g., sides or diagonals) to calculate \(k\).
- Find the ASF (\(k^2\)): Square the value of \(k\).
- Calculate the Unknown Area: Multiply the known area by \(k^2\).
✎ Common Mistake Alert!
If you are given the area ratio first (e.g., Area A is 9 times Area B), you must remember to square root this ratio to find the LSF, \(k\).
If \(\text{Area Ratio} = 9\), then \( k = \sqrt{9} = 3 \). You use \(k=3\) for calculating lengths!
Key Takeaway: Lengths use \(k\), but Areas use \(k^2\).
4. Similarity and Volume (Extended Content - E5.3)
When dealing with similar 3D solids (like cubes, prisms, or cones), we introduce the Volume Scale Factor (VSF).
If the Length Scale Factor is \(k\), the Volume Scale Factor is \(k^3\).
The Volume Relationship:
\[ \frac{\text{Volume of New Solid}}{\text{Volume of Original Solid}} = k^3 \]
Did you know? This relationship is why large animals need much thicker bones than small animals! If an animal doubles in height (\(k=2\)), its volume (and weight) increases by \(2^3=8\). Its cross-sectional bone area only increases by \(2^2=4\). The bones must compensate for the much greater volume!
The Golden Rule of Scale Factors (MUST MEMORISE!)
👉 Quick Review: The Scale Factor Pyramid
Let \(k\) be the Length Scale Factor.
- Lengths (1 dimension, e.g., cm): \( \text{Ratio} = k \)
- Areas/Surface Areas (2 dimensions, e.g., cm²): \( \text{Ratio} = k^2 \)
- Volumes (3 dimensions, e.g., cm³): \( \text{Ratio} = k^3 \)
To move DOWN the pyramid (from Length to Volume): You raise \(k\) to the power of the dimension (1, 2, or 3).
To move UP the pyramid (e.g., from Area back to Length): You use the appropriate root (e.g., \( k = \sqrt{\text{Area Ratio}} \) or \( k = \sqrt[3]{\text{Volume Ratio}} \)).
Key Takeaway: The power of the scale factor matches the dimension of the quantity you are calculating (length=1, area=2, volume=3).
5. Proving Triangles are Similar (Extended Content - E5.3)
Sometimes you need to show that two triangles are similar using geometric reasoning, rather than just being told they are similar.
The easiest and most common method is the Angle-Angle (AA) Criterion.
Method: Angle-Angle (AA)
If two triangles have two pairs of corresponding angles equal, then the triangles are similar. (Since the angles in a triangle sum to 180°, if two angles are equal, the third must also be equal.)
This is often used in situations where triangles are nested (one inside the other) or when lines are parallel.
Example Explanation:
Consider a large triangle ABC, and a smaller triangle ADE nested within it, where DE is parallel to BC.
- Angle A: Angle DAC is common to both triangle ABC and triangle ADE. (\(\angle DAE = \angle BAC\))
- Corresponding Angles: Because DE is parallel to BC, the corresponding angles are equal. (\(\angle ADE = \angle ABC\))
Since two pairs of corresponding angles are equal, triangle ADE is similar to triangle ABC (by AA similarity).
When answering exam questions, you must state the geometric reason clearly (e.g., "Common angle", "Alternate angles", or "Corresponding angles").
Key Takeaway: To prove similarity, aim to show that at least two pairs of corresponding angles are equal.
6. Putting it all together: Problem Solving Flowchart
When tackling a similarity problem, follow these steps to avoid mistakes:
- Confirm Similarity: Are the shapes similar? (Look for parallel lines, or check if angles are equal, or if the question states they are similar).
- Focus on the Scale Factor \(k\): Use the two known corresponding *lengths* (L1 and L2) to calculate the LSF: \( k = \frac{L_{\text{new}}}{L_{\text{original}}} \).
- Determine the Required Ratio:
- Need Lengths? Use \(k\).
- Need Areas/Surface Areas? Use \(k^2\).
- Need Volumes? Use \(k^3\).
- Calculate: Multiply the known value by the appropriate scale factor (\(k\), \(k^2\), or \(k^3\)) to find the unknown value.
❌ Pitfall to Avoid!
If you are given volumes (VSF) and asked to find a length, remember to work backwards:
1. Find VSF: \(\frac{V_{\text{New}}}{V_{\text{Original}}}\)
2. Find LSF (\(k\)): \( k = \sqrt[3]{\text{VSF}} \)
3. Use \(k\) to find the unknown length.
Final Key Takeaway: Similarity is defined by equal angles and proportional sides. Your entire calculation hinges on finding the Length Scale Factor, \(k\), and then applying \(k^2\) for areas and \(k^3\) for volumes.