Enlargement and Similar Figures
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- Enlargement and Similar Figures - Video - Finding Missing Length on Similar Figures using Scale Factor Watch
Theory
An enlargement multiplies every length by the same scale factor \(k\) and keeps every angle the same, so the image is similar to the original. This Year 9 maths page covers what stays the same and what changes under an enlargement: scale factors that enlarge or reduce, similarity statements, unknown sides and perimeters, and why areas change by \(k^2\) and volumes by \(k^3\).
An enlargement changes the size of a shape but not its shape. Every length is multiplied by the same number, the scale factor \(k\): \(k = \dfrac{\text{image length}}{\text{original length}}\). If \(k > 1\) the image is larger; if \(0 < k < 1\) it is a reduction.
What stays the same: every angle, the shape, and the ratio of any two sides of the figure. What changes: every length and the perimeter are multiplied by \(k\), every area by \(k^2\), and the volume of a solid by \(k^3\).
Two figures are similar if one is an enlargement or reduction of the other (it may also be turned or flipped). Matching angles are equal, and matching sides are all in the same ratio.
The symbol \(\sim\) means "is similar to" (some books write \(|\!|\!|\)). In \(ABCD \sim WXYZ\) the vertices are in matching order, so \(A \leftrightarrow W\), \(B \leftrightarrow X\), and side \(BC\) matches side \(XY\).
Congruent figures are the special case \(k = 1\): same shape and same size.
Every enlargement has a centre of enlargement. Each image point lies on the line from the centre through the original point, \(k\) times as far from the centre.
Scale factor from the original to the image:
| Under an enlargement by \(k\) | Changes to |
|---|---|
| angles | stay the same |
| lengths and perimeter | \(\times\, k\) |
| area (and surface area) | \(\times\, k^2\) |
| volume (and capacity) | \(\times\, k^3\) |
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.
Are two figures similar?
- Check that matching angles are equal.
- Divide each side of one figure by its matching side in the other. Similar figures give the same answer every time.
- Equal angles alone are not enough for shapes with more than three sides: a square and a long rectangle both have four right angles.
Areas and volumes
- Get \(k\) from a pair of matching lengths (sides, heights, radii).
- Square \(k\) for an area; cube \(k\) for a volume, capacity or mass of a solid made of the same material.
- Multiply the known area or volume, the right way up: bigger figure, bigger answer.
(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.
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: \(10 + 15 + 20 = 45\).
(i) Find the dimensions of the image.
(ii) What happens to the angles?
(iii) Compare the perimeters.
(iv) Compare the areas. By what number is the area multiplied?
| \(4 \times 1.5\) | \(=\) | \(6\text{ cm}\) |
| \(6 \times 1.5\) | \(=\) | \(9\text{ cm}\) |
(i) The image is \(6\) cm by \(9\) cm.
(ii) The angles do not change: all four are still \(90^\circ\).
(iii) Perimeters:
| \(P_{\text{original}}\) | \(=\) | \(2(4 + 6) = 20\text{ cm}\) |
| \(P_{\text{image}}\) | \(=\) | \(2(6 + 9) = 30\text{ cm}\) |
\(30 \div 20 = 1.5\), so the perimeter is multiplied by \(k\).
(iv) Areas:
| \(A_{\text{original}}\) | \(=\) | \(4 \times 6 = 24\text{ cm}^2\) |
| \(A_{\text{image}}\) | \(=\) | \(6 \times 9 = 54\text{ cm}^2\) |
| \(54 \div 24\) | \(=\) | \(\mathbf{2.25} = 1.5^2\) |
The area is multiplied by \(k^2\), not by \(k\).
(i) How much does the smaller jar hold?
(ii) The label on the smaller jar has an area of \(96\) cm\(^2\). Find the area of the matching label on the larger jar.
Lengths: \(8 : 12 = 2 : 3\). Volumes use the cube, areas use the square.
| \(\text{(i) volume}\) | \(=\) | \(810 \times \left(\dfrac{2}{3}\right)^3 = 810 \times \dfrac{8}{27}\) |
| \(\) | \(=\) | \(\mathbf{240}\text{ mL}\) |
| \(\text{(ii) area}\) | \(=\) | \(96 \times \left(\dfrac{3}{2}\right)^2 = 96 \times \dfrac{9}{4}\) |
| \(\) | \(=\) | \(\mathbf{216}\text{ cm}^2\) |
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.
What stays the same in an enlargement?
The angles, the shape and the ratio of any two sides stay the same. Every length is multiplied by the scale factor \(k\), areas by \(k^2\) and volumes by \(k^3\).
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.
If the sides are doubled, what happens to the area?
The area is multiplied by \(2^2 = 4\). The volume of a solid would be multiplied by \(2^3 = 8\).
What is the ratio of volumes of similar solids?
If matching lengths are in the ratio \(a : b\), volumes are in the ratio \(a^3 : b^3\).