Similar Figures
Every question with a fully worked solution.
- Similar Figures - Video - Finding Missing Length on Similar Figures using Scale Factor Watch
Theory
Similar figures have the same shape but not necessarily the same size. This Year 10 maths page revises how to find a scale factor, use it to find unknown sides and perimeters, decide whether two figures are similar, write a similarity statement in matching order, and work with scale drawings, plans and maps.
Two figures are similar if one is an enlargement or reduction of the other. Matching angles are equal, and matching sides are all in the same ratio.
The scale factor \(k\) is the number every length is multiplied by: \(k = \dfrac{\text{image length}}{\text{original length}}\). If \(k > 1\) the figure is enlarged; if \(0 < k < 1\) it is reduced.
The symbol \(|||\) means "is similar to". In \(ABCD \,|||\, WXYZ\) the vertices are in matching order, so \(A \leftrightarrow W\), \(B \leftrightarrow X\), and side \(BC\) matches side \(XY\).
A scale drawing, plan, model or map is similar to the real object. Its scale \(1 : n\) means \(1\) unit on the drawing stands for \(n\) of the same units in real life.
The perimeter of a similar figure is also multiplied by \(k\). Area is not: work out each length first, then the area.
Scale factor from the original to the image:
| Scale \(1 : n\) | Rule |
|---|---|
| drawing \(\to\) real | multiply by \(n\), then change units |
| real \(\to\) drawing | change to the drawing unit (usually cm), then divide by \(n\) |
| finding \(n\) | same units, then real \(\div\) drawing |
| units | \(1\text{ m} = 100\text{ cm}\), \(1\text{ km} = 1000\text{ m} = 100\,000\text{ cm}\) |
How to find an unknown side in similar figures
- Match the sides, using the similarity statement, the angles or the position in the diagram. Turned or flipped figures need care.
- Find the scale factor from one pair of matching sides where both lengths are known.
- Multiply by \(k\) to go to the image, or divide by \(k\) to go back to the original.
- Check that the larger figure has the larger answer.
How to decide whether figures are similar
- Order the sides of each figure from shortest to longest (or use the angles to match them).
- Divide each side of one figure by the matching side of the other.
- Compare: if every ratio is the same (and the angles match), the figures are similar. One different ratio means they are not.
How to use a scale \(1 : n\)
- Decide the direction: drawing to real (multiply) or real to drawing (divide).
- Keep the same unit until the multiplying or dividing is done.
- Convert to the unit asked for, such as m or km.
(i) the scale factor
(ii) \(XY\)
(iii) \(DA\)
The letters match in order: \(AB \leftrightarrow WX\), \(BC \leftrightarrow XY\), \(DA \leftrightarrow ZW\).
| \(k\) | \(=\) | \(\dfrac{\text{image}}{\text{original}} = \dfrac{WX}{AB} = \dfrac{7}{5} = \mathbf{1.4}\) |
| \(XY\) | \(=\) | \(6 \times 1.4 = \mathbf{8.4}\text{ cm}\) |
| \(DA\) | \(=\) | \(9.8 \div 1.4 = \mathbf{7}\text{ cm}\) |
Going to the larger figure, multiply. Going back to the smaller figure, divide.
(i) A track measures \(9.2\) cm on the map. How long is the real track, in km?
(ii) Two huts are \(1.5\) km apart. How far apart are they on the map?
(i) Map to real: multiply by \(25\,000\), then change units.
| \(9.2 \times 25\,000\) | \(=\) | \(230\,000\text{ cm}\) |
| \(\) | \(=\) | \(2300\text{ m}\) |
| \(\) | \(=\) | \(\mathbf{2.3}\text{ km}\) |
(ii) Real to map: change to cm first, then divide by \(25\,000\).
| \(1.5\text{ km}\) | \(=\) | \(150\,000\text{ cm}\) |
| \(150\,000 \div 25\,000\) | \(=\) | \(\mathbf{6}\text{ cm}\) |
(i) Write the scale as \(1 : n\).
(ii) A doorway is \(0.6\) cm wide on the plan. How wide is the real doorway?
(i) Write both lengths in the same unit, then divide by the plan length.
| \(12\text{ m}\) | \(=\) | \(1200\text{ cm}\) |
| \(8 : 1200\) | \(=\) | \(1 : 150\) |
| \(\text{scale}\) | \(=\) | \(\mathbf{1 : 150}\) |
(ii) Plan to real: multiply.
| \(0.6 \times 150\) | \(=\) | \(\mathbf{90}\text{ cm} = 0.9\text{ m}\) |
Every side is multiplied by \(k\), so the perimeter is multiplied by \(k\) too.
| \(P_{\text{small}}\) | \(=\) | \(4 + 6 + 8 = 18\text{ cm}\) |
| \(k\) | \(=\) | \(\dfrac{45}{18} = 2.5\) |
| \(\text{sides}\) | \(=\) | \(4 \times 2.5,\ 6 \times 2.5,\ 8 \times 2.5\) |
| \(\) | \(=\) | \(\mathbf{10},\ \mathbf{15},\ \mathbf{20}\text{ cm}\) |
Check: \(10 + 15 + 20 = 45\).
Common pitfalls
Frequently asked questions
What makes two shapes similar?
Two shapes are similar if their matching angles are equal and their matching sides are all in the same ratio. One is an enlargement or reduction of the other.
How do you find the scale factor of similar figures?
Divide a length on the image by the matching length on the original. If a \(5\) cm side becomes \(7\) cm, \(k = \dfrac{7}{5} = 1.4\).
How do you find a missing side of a similar shape?
Find the scale factor from a pair of matching sides, then multiply the matching side by it to go to the larger figure, or divide by it to go to the smaller figure.
What does a scale of 1 to 25000 mean?
One unit on the map stands for \(25\,000\) of the same units in real life. \(1\) cm on the map is \(25\,000\) cm, which is \(250\) m.
Does the perimeter change by the scale factor?
Yes. Every side is multiplied by the scale factor, so the perimeter is too. Area is different, so find the new lengths before working out an area.