Scale Factors Diagrams and Maps for ESAT

Similar Shapes and Scale Factor

When two shapes are mathematically similar, the lengths of the sides of one shape are directly proportional to the lengths of the corresponding sides of the other. We can determine the side lengths of one shape from another by multiplying each original length by a constant known as the scale factor.

Consider the diagram below showing three similar triangles. The first triangle serves as the base with side lengths $x$, $y$, and $z$. The other two triangles are related to it by constants $a$ and $b$, which are the scale factors.

A scale factor can be greater than 1 or less than 1. In this specific diagram, $a > 1$, indicating an enlargement, while $b < 1$, indicating a reduction.

Enlargements and Scale Factor Calculations

In an enlargement, every dimension is increased by the same multiplier. We can see this in the relationship between Trapezium A and Trapezium B, where Trapezium B is an enlargement of Trapezium A with a scale factor of 3.

To find the missing values $p$, $q$, $r$, and $s$ on Trapezium B, we multiply each corresponding length on Trapezium A by the scale factor 3:

1. $p = 3 \times 10 = 30$
2. $q = 3 \times 4 = 12$
3. $r = 3 \times 14 = 42$
4. $s = 3 \times 3 = 9$

Scale Diagrams

A scale diagram is a representation of an object or drawing that is mathematically similar to the original, though usually smaller. In the triangle examples discussed previously, each triangle is effectively a scale drawing of the others.

When working with scale diagrams, we often need to determine the scale factor by comparing known corresponding sides. Consider the following figures where Figure B is a scale drawing of Figure A.

We can see that the base of Figure B is $7.5 \text{ cm}$, while the corresponding base of Figure A is $15 \text{ cm}$. Since $7.5 \text{ cm}$ is half of $15 \text{ cm}$, the scale factor from A to B is $\frac{1}{2}$.

To find the missing values $x$ and $y$:

1. The length $y$ corresponds to the length $10$ on Figure A. Therefore, $y = 10 \times \frac{1}{2} = 5$.
2. The length $2.5$ on Figure B corresponds to the length $x$ on Figure A. Since $x \times \frac{1}{2} = 2.5$, we find $x$ by multiplying $2.5$ by 2, giving $x = 5$.

Maps and Ratio Scales

A map is a specific type of scale diagram. Its scale is frequently expressed as a ratio. For instance, if $1 \text{ cm}$ on a map represents $1 \text{ km}$ in reality, we can write the scale as $1 \text{ cm}$ represents $1 \text{ km}$ or as a unitless ratio $1 : 100,000$. This is because $1 \text{ km} = 1,000 \text{ m} = 100,000 \text{ cm}$.

Worked Example: Map to Real Life

A map of a village uses a scale of $1 : 25,000$. If the distance from a bank to a post office on the map is $3 \text{ cm}$, what is the actual distance in metres?

1. First, calculate the distance in centimetres: $3 \text{ cm} \times 25,000 = 75,000 \text{ cm}$.
2. Next, convert centimetres to metres. Since there are $100 \text{ cm}$ in a metre: $75,000 \div 100 = 750 \text{ m}$.

Worked Example: Real Life to Map

Using the same map scale of $1 : 25,000$, if the actual distance between a supermarket and a bank is $200 \text{ metres}$, what is this distance in millimetres on the map?

1. First, convert the real distance to centimetres: $200 \text{ m} = 20,000 \text{ cm}$.
2. Divide the real distance by the scale factor to find the map distance: $20,000 \div 25,000 = 0.8 \text{ cm}$.
3. Finally, convert centimetres to millimetres. Since there are $10 \text{ mm}$ in $1 \text{ cm}$: $0.8 \text{ cm} \times 10 = 8 \text{ mm}$.