Hello IGCSE 0607 Mathematician!

Welcome to the chapter on Similarity! This is one of the most practical and useful topics in Geometry. It allows us to relate the size of big things to small models, making calculations involving maps, architecture, and even complex volumes surprisingly straightforward.

In short, similarity is about things that have the same shape but different sizes. Think of zooming in or out on a digital photo—the proportions stay exactly the same. Ready to master the magic of scaling?

1. Understanding Similar Shapes

1.1 What does "Similar" mean?

Two shapes are mathematically similar if one is an exact enlargement or reduction of the other. Crucially, similarity preserves shape, but not size.

Imagine holding a tiny toy car next to a real car. If the toy car is perfectly scaled, they are similar shapes.

The Two Golden Rules of Similarity:
  1. Corresponding Angles are Equal: This is non-negotiable! If two shapes are similar, their matching corners must have the exact same angle.
  2. Ratio of Corresponding Sides is Constant: If you divide the length of one side of the larger shape by the length of the corresponding side of the smaller shape, you always get the same number. This number is called the Scale Factor (\(k\)).
Quick Review: Similarity vs. Congruence

Don't worry if this sounds confusing!

  • Congruent Shapes: Same shape, same size (identical twins).
  • Similar Shapes: Same shape, different size (older brother and younger sister wearing the same shirt design).

1.2 Finding the Linear Scale Factor (\(k\))

The scale factor (\(k\)) is the key to solving similarity problems involving lengths.

We define \(k\) based on the direction of the enlargement/reduction:

\(k = \frac{\text{Length on New Shape (Image)}}{\text{Length on Old Shape (Object)}}\)

  • If \(k > 1\), the shape has been enlarged.
  • If \(0 < k < 1\), the shape has been reduced.

Step-by-Step: Calculating an Unknown Length

  1. Identify corresponding known sides on both shapes (e.g., the shortest side on the new shape corresponds to the shortest side on the old shape).
  2. Calculate the Linear Scale Factor (\(k\)): Divide the corresponding known length of the target shape by the length of the original shape.
  3. Find the Unknown Length: Multiply the corresponding known side of the original shape by \(k\).
    \(\text{Unknown Length} = k \times \text{Known Corresponding Length}\)
Key Takeaway (Lengths):

For similar shapes, all corresponding lengths (sides, perimeter, height, radius) use the same linear scale factor, \(k\).

2. Proving Similarity in Triangles (Extended Focus)

Triangles are the most common shapes in similarity questions. Because of their rigid structure, we only need three criteria to prove they are similar. The syllabus requires you to show similarity using geometric reasons.

2.1 Criteria for Similar Triangles

1. Angle-Angle (AA) or Angle-Angle-Angle (AAA)

If two pairs of corresponding angles are equal, the triangles are similar. (Since the angles in a triangle always sum to \(180^{\circ}\), knowing two means the third must also be equal.)

This is the fastest and easiest way to prove similarity!

2. Side-Side-Side (SSS) Ratio

If the ratios of all three pairs of corresponding sides are equal (i.e., they all share the same scale factor \(k\)), the triangles are similar.

3. Side-Angle-Side (SAS) Ratio

If two pairs of corresponding sides have the same ratio, AND the included angle (the angle between those two sides) is equal, the triangles are similar.

Common Mistake to Avoid:

When using SSS or SAS, make sure you match the sides correctly! Always pair the shortest side of Triangle A with the shortest side of Triangle B, the longest with the longest, and so on.

2.2 Similarity in Parallel Lines

A common setup for similarity involves a smaller triangle sitting inside a larger one, often created by parallel lines (like a triangle cut by a line segment parallel to its base).

When you have a diagram like this, you can immediately use the AA criterion, because of the properties of parallel lines:

  • The angle at the common vertex (usually the top) is shared by both triangles.
  • The corresponding angles created by the parallel line cutting across the transversal sides are equal.

Because their angles are equal, the two triangles must be similar.

Key Takeaway (Proving Similarity):

For triangles, if you can show AA (two angles are equal), you have proven similarity. Use properties like parallel lines, vertically opposite angles, or angles on a straight line to find equal angles.

3. The Relationship between Length, Area, and Volume (Extended Focus)

This section is crucial for solving problems involving 3D solids or calculating paint requirements (Area) versus capacity (Volume).

If Shape A is similar to Shape B with a Linear Scale Factor \(k\), then the ratios of their areas and volumes follow specific rules.

3.1 The Area Rule (\(k^2\))

The ratio of the areas of two similar shapes is equal to the square of the linear scale factor.

If \(\frac{\text{Length B}}{\text{Length A}} = k\)

Then \(\frac{\text{Area B}}{\text{Area A}} = k^2\)

Did you know?
The reason we square the scale factor is simple: Area is calculated by multiplying two lengths together (e.g., length × width). Since both the length and the width are scaled by \(k\), the total area scales by \(k \times k = k^2\).

3.2 The Volume Rule (\(k^3\))

The ratio of the volumes of two similar solids is equal to the cube of the linear scale factor.

If \(\frac{\text{Length B}}{\text{Length A}} = k\)

Then \(\frac{\text{Volume B}}{\text{Volume A}} = k^3\)

Analogy: The Paint and Water Test

Imagine a small cube (Side = 1m) and a similar large cube (Side = 2m).
The linear scale factor is \(k = 2\).

  • Length: The large cube is 2 times bigger (k=2).
  • Area (Surface Area/Paint needed): The large cube has a surface area 4 times bigger (\(k^2 = 2^2 = 4\)).
  • Volume (Capacity/Water needed): The large cube holds a volume 8 times bigger (\(k^3 = 2^3 = 8\)).

3.3 Summary of Similarity Ratios

Ratio Type Relationship to Linear Scale Factor (\(k\))
Length Ratio (Side, Height, Perimeter) \(k\)
Area Ratio (Area, Surface Area) \(k^2\)
Volume Ratio (Volume, Capacity) \(k^3\)

3.4 Working Backwards: Finding \(k\) from Area or Volume

Sometimes you are given the area or volume ratio first and need to find the linear scale factor \(k\).

  • To find \(k\) from an area ratio, you take the square root:
    \(\text{Area Ratio} = k^2 \Rightarrow k = \sqrt{\text{Area Ratio}}\)
  • To find \(k\) from a volume ratio, you take the cube root:
    \(\text{Volume Ratio} = k^3 \Rightarrow k = \sqrt[3]{\text{Volume Ratio}}\)

Example Process:

Two similar cones, A and B, have volumes of \(27 \text{ cm}^3\) and \(125 \text{ cm}^3\), respectively. Find the ratio of their heights.

  1. Find Volume Ratio: \(\frac{\text{Volume B}}{\text{Volume A}} = \frac{125}{27}\).
  2. Find Linear Scale Factor (\(k\)): Since this is a volume ratio, we take the cube root.
    \(k = \sqrt[3]{\frac{125}{27}} = \frac{5}{3}\)
  3. Answer the Question: The ratio of their heights (a length) is \(k\).
    Therefore, \(\frac{\text{Height B}}{\text{Height A}} = \frac{5}{3}\) (or 5:3).
Key Takeaway (Area and Volume):

Remember the simple dimensional trick: Length is 1D (\(k\)), Area is 2D (\(k^2\)), and Volume is 3D (\(k^3\)). Know how to move between these dimensions using roots and powers!