Lengths in Similar Shapes (4.4.1) | CIE IGCSE Maths

Understanding the concept of similarity and scale factors is crucial in geometry, particularly when we deal with similar shapes. Similar shapes are identical in form but differ in size. This discrepancy in size can be quantified using a scale factor, which provides a multiplier for transforming one shape into another.

What is a Scale Factor?

A scale factor is a number which scales, or multiplies, some quantity. In the context of similar shapes, the scale factor between two similar shapes is the ratio of lengths of any two corresponding sides. For example, if the scale factor is 3, every length in the similar shape is 3 times longer than the corresponding length in the original shape.

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Key Points:

Definition: A multiplier that determines how much one shape has been scaled to become another similar shape.
Identification: Found by comparing corresponding lengths of two similar shapes.
Application: Used to calculate unknown lengths, areas, and volumes in similar shapes.

Calculating Lengths Using Scale Factors

When dealing with similar shapes, calculating lengths, areas, and volumes can be straightforward if we understand how to apply scale factors properly.

Example 1: Calculating Lengths

Question: Given two similar rectangles, Rectangle A and Rectangle B, where the length of Rectangle A is 4 cm and the scale factor from Rectangle A to Rectangle B is 3, find the length of Rectangle B.

Solution:

1. Identify the scale factor: The scale factor from Rectangle A to Rectangle B is 3.

2. Apply the scale factor:

\text{ Length of Rectangle B } = \text{ Length of Rectangle A × Scale Factor }

\text{ Length of Reactangle B } = 4 \text{ cm } × 3 = 12 \text{ cm }

3. Conclusion: The length of Rectangle B is 12 cm.

Example 2: Using Scale Factors in Triangles

Question: A triangle has sides of length 5 cm, 7 cm, and 9 cm. A similar triangle has a longest side of 27 cm. What is the scale factor between the two triangles, and what are the lengths of the other sides in the larger triangle?

Solution:

1. Find the scale factor:

Scale Factor = Length of the longest side in the larger triangle ÷ Length of the longest side in the original triangle

\text{ Scale Factor } = 27 \text{ cm } ÷ 9 \text{ cm } = 3

2. Calculate the lengths: Apply the scale factor to the other sides.

\text{ Side } 1: 5 \text{ cm } × 3 = 15 \text{ cm }

\text{ Side } 2: 7 \text{ cm } × 3 = 21 \text{ cm }

3. Conclusion: The lengths of the other sides in the larger triangle are 15 cm and 21 cm, respectively.

Practice Questions

Question 1: Simple Scaling

If the side of a square is increased by a scale factor of 2, what is the new length of the side?

Solution:

\text{ Simple Scaling Solution }: \text{ New length } = \text{ Original length } × \text{ Scale factor } = \text{ Side } \times{ 2 }

Question 2: Complex Figures

A parallelogram has sides 10 cm and 15 cm. A similar parallelogram has a smaller side measuring 20 cm. Find the length of the larger side in the similar parallelogram.

Solution:

Identify the scale factor:

\text{ Scale Factor } = \text{ New smaller side } ÷ \text{ Original smaller side }

\text{ Scale Factor }= 20 \text{ cm } ÷ 10 \text{ cm } = 2

Apply the scale factor to the larger side:

\text{ New larger side } = \text{ Original larger side } × \text{ Scale factor }

\text{ New larger side } = 15 \text{ cm } \times 2 = 30 \text{ cm }

Written by:

Dr Rahil Sachak-Patwa

Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

Oxford University - PhD Mathematics

CIE IGCSE Maths Study Notes

4.4.1 Lengths in Similar Shapes

What is a Scale Factor?

Key Points:

Calculating Lengths Using Scale Factors

Example 1: Calculating Lengths

Solution:

Example 2: Using Scale Factors in Triangles

Solution:

Practice Questions

Question 1: Simple Scaling

Solution:

Question 2: Complex Figures

Solution:

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