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D) Similar Shapes: Length

D) Similar Shapes: Length

Scale factors appear in a few different sections in maths. We can use scale factors to compare different sized shapes that are mathematically similar. Mathematically similar shapes are where the respective angles in the two similar shapes are the same and the respective sides of the two similar shapes are in the same proportion.

A scale factor that is greater than 1 will increase the length (surface area and volume) of a shape and a scale factor that is less than 1 will decrease the length (surface area and volume).

A scale factor that is greater than 1 will increase the length (surface area and volume) of a shape and a scale factor that is less than 1 will decrease the length (surface area and volume).

**Example 1**

Let’s have an example. We have two rectangles that are mathematically similar. The smaller rectangle’s shortest side has a length of 5 cm and it’s longest side is 9 cm. The shortest side of the larger shape is 15 cm and it’s longest side is unknown. What is the scale factor and what is the length of the larger shape’s longest side?

The first step to answering this question is to find out what the scale factor of enlargement is. In order to work the scale factor out, we need to use the values of corresponding shapes. In the question, we are given the values for the shortest sides of both of the rectangles. The shortest side of the smaller rectangle has a length of 5 cm and the larger rectangle has a shorter length of 15 cm. We are able to work out the scale factor by dividing the larger length by the shorter length.

This tells us that the scale factor is 3. We are now able to find the length of the longer side in the larger rectangle and we do this by multiplying the longer side in the smaller shape by the scale factor that we have just found. The longer side in the small rectangle has a length of 9 cm.

Therefore, the larger rectangle’s longest side has a length of 27cm.

**Example 2**

The two shapes below are mathematically similar. What is the length of the side labelled

*x*?

The first step in answering this question is to find out what the scale factor is. We find the scale factor by dividing two of the similar sides. We are given the base in both the larger and the smaller shape, therefore we are going to work out the scale factor from these two sides. The easiest way to work out the scale factor is to always divide the longer side by the smaller side. The longer side is 12 cm and the shorter side is 3 cm.

The scale factor for these two similar shapes is 4. This is the scale factor that we use to find the length of the respective side in the larger shape from the length of the respective side in the smaller shape. However, we are given the length of the longer side and we want to find the length of the respective side in the smaller shape. This means that we do the opposite of multiplying by 4, and instead divide by 4. Therefore, we are going to divide 16 cm by 4.

The length of the side labelled

It is always a good idea to check that our answer makes sense and we do this by checking that the lengths of all of the sides that we are given in the smaller shape are smaller than the length of their respective sides in the larger shapes. This is the case in the question above, which means that we have used correct operation (multiply or divide) for the scale factor.

*x*is 4 cm.It is always a good idea to check that our answer makes sense and we do this by checking that the lengths of all of the sides that we are given in the smaller shape are smaller than the length of their respective sides in the larger shapes. This is the case in the question above, which means that we have used correct operation (multiply or divide) for the scale factor.