Similar Lengths (Edexcel IGCSE Maths)

Spec point — 4.11A

Similar lengths

How do I find the scale factor between lengths on similar shapes?

• Equivalent lengths on two similar shapes will be in the same ratio and are linked by a scale factor

• Establish the type of enlargement

  • If the second shape is bigger

    • then the scale factor is greater than 1

  • If the second shape is smaller

    • then the scale factor is greater than 0 but less than 1

• To find the scale factor

  • Identify **known **lengths of corresponding sides
  • Divide the**** length on the second shape by the corresponding length on the first shape

How do I find missing lengths on similar shapes?

Method 1

• STEP 1 Find the scale factor to get from the first shape to the second shape

  • Divide a length on the second by the corresponding length on the first
  • The scale factor can be less than 1 for this method

• STEP 2 Use the scale factor to find the length you need

  • To find a missing length on the second shape

    • Multiply the corresponding length on the first shape by the scale factor

  • To find a missing length on the first shape

    • Divide the corresponding length on the first shape by the scale factor

Method 2

• STEP 1 Find the scale factor to get from the smaller shape to the bigger shape

  • Divide a length on the bigger shape by the corresponding length on the smaller shape
  • The scale factor is always greater than 1 for this method

• STEP 2 Use the scale factor to find the length you need

  • To find a missing length on the bigger shape

    • Multiply the corresponding length on the smaller shape by the scale factor

  • To find a missing length on the smaller shape

    • Divide the corresponding length on the bigger shape by the scale factor

Exam Hint If similar shapes overlap on the diagram (or are not clear) draw them separately.

For example, in this diagram the triangles ABC and APQ are similar:

So redraw them separately before starting:

Worked Example ABCD and PQRS are similar shapes. Find the length of PS.

Answer:

The two shapes are mathematically similar

Identify two known corresponding sides of the similar shapes

AB and PQ are corresponding sides*

Method 1

The second shape is smaller than the first shape so the scale factor will be between 0 and 1 Divide the known length on the second shape by the corresponding length on the first shape to find the scale factor

$\begin{gathered} \text{Scale} \text{Factor} & = \frac{3}{6} = 1 \frac{1}{2} \end{gathered}$

Multiply the length AD by the scale factor to find its corresponding length PS on the second shape

$\begin{array}{rcl} P S & = 1 \frac{1}{2} \times 15 = \frac{15}{2} \\ \end{array}$

$PS=7.5\mathrm{cm}$

Method 2

Divide the known length on the bigger shape by the corresponding length on the smaller shape to find the scale factor

$\begin{gathered} \text{Scale} \text{Factor} & = \frac{6}{3} = 2 \end{gathered}$

The missing length is on the smaller shape

Divide the length AD by the scale factor to find its corresponding length PS on the smaller shape

$PS=15\div2$

$PS=7.5\mathrm{cm}$