This module builds on: Module 5 (M5) , Module 6 (M6) and Module 7 (M7) transformations.
Similar shapes
When a shape is enlarged, the image is similar to the original shape. It is the same shape but a different size.
These two shapes are similar as they are both rectangles but one is an enlargement of the other.
Similar Triangles
Two triangles are similar if the angles are the same size or the corresponding sides are in the same ratio. Either of these conditions will prove two triangles are similar.
Triangle B is an enlargement of triangle A by a The ratio of corresponding lengths in similar shapes, ie how much larger or smaller the shapes are. of 2. Each length in triangle B is twice as long as in triangle A.
The two triangles are similar.
Example 1
State whether the two triangles are similar. Give a reason to support your answer.
Answer: Yes, they are similar. The two lengths have been increased by a scale factor of 2. The corresponding angle is the same.
Example 2
State whether the two triangles are similar. Give a reason to support your answer.
To decide whether the two triangles are similar, calculate the missing angles.
Remember angles in a triangle add up to 180°.
Angle yxz = \(180 - 85 - 40 = 55^\circ\)
Angle YZX = \(180 - 85 - 55 = 40^\circ\)
Answer: Yes, they are similar. The three angles are the same.
Example 3
State whether the two triangles are similar. Give a reason to support your answer.
Answer: No. Two of the sides of the triangle are increased by a scale factor of 1.5. The other side has been increased by a scale factor of 2.
Example
Calculate the length of side
(i) AC;
(ii) DB.
Solution
Triangles ABC and BDE are similar because:
- They share the angle B.
- Angle A = angle D because lines AC and DE are parallel.
- Angle C = Angle E because lines AC and DE are parallel.
To calculate missing lengths, begin by drawing the triangles out separately.
The scale factor of enlargement can be found by dividing the length of BC by BE.
Scale factor \(= 7.2 \div 4.8 = 1.5\)
(i)
\(AC = DE \times 1.5\)
\(AC = 5 \times 1.5 = 7.5 \text{cm}\)
(ii)
\(AB = 1.5 \times DB\)
\(DB = AB \div 1.5\)
\(DB = 5.4 \div 1.5 = 3.6 \text{cm}\)
Question
Find the value of \(x\).
Answer
\(\text{Scale factor} = 24 \div 20 = 1.2\)
\(x = 24 \times 1.2 = 28.8 \text{mm}\)
Enlargement Effect on Perimeter, Area and Volume
When we enlarge a shape by a scale factor, the length of each edge and the perimeter are multiplied by the scale factor.
When we enlarge a shape by a scale factor, the area of the shape is multiplied by the square of the scale factor.
When we enlarge a shape by a scale factor, the volume of the shape is multiplied by the cube of the scale factor.
Example
Two similar posters have perimeters of \(20 \text{cm}\) and \(80 \text{cm}\). The area of the smaller poster is \(24 \text{cm}^{2}\). Calculate the area of the larger poster.
Solution
The scale factor of the perimeters is \(80 \div 20 = 4\)
Perimeter is a measurement of length.
The scale factor of the areas will be \(4^2 = 16\)
The area of the larger poster will be \(24 \times 16 = 384 \text{cm}^{2}\).
Ratios
Ratios can also be used to express scale factor.
Example
The area of triangle A is \(216 \text{cm}^{2}\). The ratio of the side lengths of triangle A to triangle B is 3:1.
Calculate the area of triangle B.
Solution
\(\text{length ratio} = 3:1\)
\(\text{area ratio} = 3^2:1^2 = 9:1\)
\(\text{area of A} = 216 \text{cm}^{2}\)
\(\text{area of B} = 216 \div 9 = 24 \text{cm}^{2}\)
Example
A photo has dimensions \(8 \text{cm} \times 10 \text{cm}\).
An enlargement of this photo has an area of \(320 \text{cm}^{2}\).
Find the ratio of the areas and the hence the dimensions of the enlarged photo.
Solution
Area of first photo is \(80 \text{cm}^{2}\).
Ratio of areas \(80 : 320 = 1:4\)
Ratio of lengths \(\sqrt{1} : \sqrt{4} = 1 : 2\)
Dimensions of second photo \(8 \times 2 = 16\)
\(10 \times 2 = 20\)
Dimensions of the enlarged photo are \(16 \text{cm} \times 20 \text{cm}\).
Test yourself
More on M7: Geometry and measures
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