Properties of Similar Figures
The scale factor between similar figures is the number every length is multiplied by, found as a new length divided by the matching old length. This NSW Year 9 Core maths subtopic uses it to find unknown sides, and is Stage 5 Core content under NESA outcome MA5‑GEO‑C‑01.
You will find scale factors for enlargements and reductions, match sides correctly even when one figure has been turned, and find unknown sides, as when a \(5\) cm by \(7\) cm rectangle is enlarged so the \(5\) cm side becomes \(15\) cm. You will apply scale factors to circles, find sides in similar triangles formed by a parallel line or a bow-tie shape, resize a photograph, find a flagpole's height from its shadow, and find the perimeter of a scaled triangle. It is part of Year 9 Core Congruence and Similar Figures.
Every question with a fully worked solution.
- Properties of Similar Figures - Video - Similar Polygons Watch
Theory
The scale factor is the number every length of a figure is multiplied by to give a similar figure. This Year 9 maths page shows how to find the scale factor from a pair of matching sides, use it to find an unknown side, and spot similar triangles made by a parallel line, two triangles meeting point to point, and shadows.
When a figure is enlarged or reduced, every length is multiplied by the same number. That number is the scale factor, \(k\).
Find \(k\) from a pair of matching sides whose lengths you both know. Divide the length on the new figure by its partner on the old figure: new over old.
If \(k > 1\) the figure is an enlargement. If \(k < 1\) it is a reduction. If \(k = 1\) the figures are congruent.
Angles do not change. Only lengths are multiplied by \(k\), and that includes the perimeter, the radius, the diameter and the circumference.
Similar triangles often hide inside a diagram. A line parallel to one side of a triangle cuts off a smaller similar triangle. Two triangles meeting point to point, with parallel outer sides, are similar too.
| What you want | What to do |
|---|---|
| the scale factor | divide a new length by its old partner |
| a length on the new figure | multiply the old length by \(k\) |
| a length on the original | divide the new length by \(k\) |
How to find an unknown side in similar figures
- Match the figures: use the similarity statement, or longest with longest and shortest with shortest.
- Choose a pair of matching sides with both lengths known.
- Divide to get \(k\), new over old. Check it is bigger than \(1\) for an enlargement and smaller for a reduction.
- Multiply the old side by \(k\) to get the new side, or divide a new side by \(k\) to get the old side.
- Check that the new answer gives the same ratio.
Use the pair with both lengths known to find \(k\), new over old. Then multiply the other old side by \(k\).
| \(k\) | \(=\) | \(\dfrac{20}{8} = 2.5\) |
| \(\text{other side}\) | \(=\) | \(12 \times 2.5\) |
| \(\) | \(=\) | \(\mathbf{30} \text{ cm}\) |
Check: \(\dfrac{30}{12} = 2.5\), the same factor.
Match longer side with longer side.
| \(k\) | \(=\) | \(\dfrac{12}{30} = 0.4\) |
| \(\text{shorter side}\) | \(=\) | \(20 \times 0.4\) |
| \(\) | \(=\) | \(\mathbf{8} \text{ cm}\) |
The scale factor is less than \(1\), which is what a reduction should give.
\(\angle A\) is shared, and \(\angle ADE = \angle ABC\) (corresponding angles, \(DE \parallel BC\)). So \(\triangle ADE \sim \triangle ABC\). \(AD\) matches the whole side \(AB\), not just \(DB\).
| \(AB\) | \(=\) | \(4 + 6 = 10\) |
| \(k\) | \(=\) | \(\dfrac{AB}{AD} = \dfrac{10}{4} = 2.5\) |
| \(BC\) | \(=\) | \(3 \times 2.5\) |
| \(\) | \(=\) | \(\mathbf{7.5}\) |
Read the matching off the statement: \(A \leftrightarrow E\), \(B \leftrightarrow D\), \(C \leftrightarrow C\). So \(AC\) matches \(EC\), and \(BC\) matches \(DC\).
| \(k\) | \(=\) | \(\dfrac{CE}{CA} = \dfrac{4.5}{3} = 1.5\) |
| \(CD\) | \(=\) | \(4 \times 1.5 = \mathbf{6}\) |
| \(DE\) | \(=\) | \(5 \times 1.5 = \mathbf{7.5}\) |
Common pitfalls
Frequently asked questions
How do you find the scale factor of similar figures?
Pick a pair of matching sides whose lengths you both know. Divide the length on the new figure by the length on the original. For a \(4\) cm side that becomes \(10\) cm, \(k = \dfrac{10}{4} = 2.5\).
How do you find a missing side in similar shapes?
Find the scale factor from a matching pair, then multiply the matching side of the original by it. To go back from the new figure to the original, divide by the scale factor instead.
What does a scale factor less than 1 mean?
The figure has been reduced: every length on the new figure is shorter than on the original.
How do you use shadows to find the height of a tree?
Measure a post and its shadow at the same time as the tree's shadow. The triangles are similar, so multiply the post's height by the ratio of the shadows. A \(1.5\) m post with a \(2\) m shadow and a tree with a \(14\) m shadow give \(k = 7\), so the tree is \(10.5\) m tall.
Does the perimeter change by the scale factor?
Yes. Every side is multiplied by the scale factor, so the perimeter is too.