Scale Factors

A scale factor is the enlargement or reduction of a shape by a specific amount, which can be positive or negative. It allows us to compare the size of one object to another by expressing the proportional change in the dimensions of the object.

For example, an enlargement with a scale factor of 2 means that the new shape would have all its sides doubled. In contrast, an enlargement with a scale factor of \frac{1}{2} means that the new shape would have all its sides halved.

Scale factors allow us to represent objects that are proportionally larger or smaller while maintaining their original shape.

Let’s look at an example:

A 4 cm × 4 cm square enlarged by a scale factor of 4 becomes a 16 cm × 16 cm square.

Let’s look at another example:

If two triangles are similar and the ratio of their corresponding side lengths is 3:1, then the scale factor between the two triangles is 3.

To find the length of a side in the larger triangle, simply multiply the length of the corresponding side in the smaller triangle by the scale factor.

You can calculate scale factors from measurements by dividing the length of a side on the larger shape by the length of the matching side on the smaller shape. For example, if a smaller rectangle has a width of 4 cm and a larger, similar rectangle has a width of 12 cm, the scale factor from the smaller rectangle to the larger rectangle is 12 ÷ 4 = 3.

Converting Scale Factors

Converting scale factors is necessary when working with different units or when needing to resize an object. To convert scale factors, you must first identify the conversion factor between the units.

For example, if a map uses a scale factor of 1 : 50,000, it means that 1 cm on the map represents 50,000 cm in reality. If you want to convert this scale factor to kilometres, you need to apply the conversion factor between centimetres and kilometres.

Since 1 km = 100,000 cm, the conversion factor is \frac{1}{100,000}. Multiply the original scale factor by the conversion factor: \frac{1}{100,000} \times (1:50,000) = 1:0.5, so 1 cm on the map represents 0.5 km in reality.

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