Scale Factors Scale Diagrams and Maps for University Admission

Similar Shapes and Scale Factors

When two geometric shapes are mathematically similar, the lengths of the sides of one can be calculated from the corresponding sides of the other by using a constant multiplier. This multiplier is known as the scale factor. In similarity, while lengths change, the internal angles of the shapes remain identical.

In the diagram above, the three triangles are similar. The first triangle has side lengths $x$, $y$, and $z$. The constants $a$ and $b$ represent the scale factors used to generate the other triangles. Note that a scale factor can be a fraction less than 1: if $a > 1$, the shape is enlarged: if $b < 1$, the shape is reduced.

Worked Example: Enlarging Trapeziums

Trapezium B is a mathematical enlargement of Trapezium A with a scale factor of 3. We are required to find the values of the missing sides $p$, $q$, $r$, and $s$.

Because the scale factor is 3, every single length on Trapezium A must be multiplied by 3 to determine the corresponding length on Trapezium B:

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

Scale Diagrams

A scale diagram is a mathematically similar representation of an original object or drawing, usually rendered at a smaller size. By comparing a known length on both diagrams, the scale factor can be deduced and used to find other unknown dimensions.

Worked Example: Using Similarity in Scale Drawings

In the following figures, Figure B is a scale drawing of Figure A. We need to calculate the values of $x$ and $y$.

First, we identify corresponding sides with known lengths to find the scale factor. We see that the length $7.5\text{ cm}$ in Figure A corresponds to $15\text{ cm}$ in Figure B. Since $7.5$ is half of $15$, the scale factor from Figure B to Figure A is $\frac{1}{2}$. Conversely, the scale factor from Figure A to Figure B is 2.

1. To find $y$ (a length in Figure A): $y = 10 \times \frac{1}{2} = 5$.
2. To find $x$ (a length in Figure B): Since the corresponding length in Figure A is $2.5$, we know that $\frac{1}{2}x = 2.5$. Solving this gives $x = 2.5 \times 2 = 5$.

Maps

Maps are specific types of scale diagrams. The scale is typically given as a ratio $1 : n$. For instance, a scale of $1 : 100,000$ means that 1 unit on the map represents $100,000$ of the same units in real life. Because $100,000\text{ cm} = 1\text{ km}$, this could also be described as 1 cm representing 1 km.

Worked Example: Map Calculations

A map has a scale of $1 : 25,000$.

Scenario 1: The map distance from the bank to the post office is $3\text{ cm}$. How far is this in reality, in metres?

First, calculate the distance in centimetres: $3\text{ cm} \times 25,000 = 75,000\text{ cm}$. Next, convert to metres by dividing by 100: $75,000 \div 100 = 750\text{ m}$.

Scenario 2: The real distance from the supermarket to the bank is $200\text{ metres}$. How many millimetres is this on the map?

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