Edexcel GCSE Foundation Maths - What are Similar Shapes?

What are Similar Shapes?

Similar shapes are shapes that have the same angles but different sizes.

This means that their corresponding sides are in the same ratio.

Key points:

• Same angles: All corresponding angles are equal.
• Different sizes: Corresponding sides are in the same proportion (ratio).

Understanding Similarity

Think of a photograph. You can enlarge or shrink a photo, but the proportions remain the same.

Example:

• A small square and a large square are similar shapes.
• A small triangle and a large triangle can also be similar shapes, as long as their corresponding angles are equal.

Identifying Similar Shapes

To identify similar shapes, follow these steps:

1. Check for equal angles: Make sure all corresponding angles are equal.
2. Calculate the scale factor: Find the ratio between corresponding sides. This ratio should be the same for all sides.

Example:

Two triangles are similar:

• Triangle A: Sides of length 3 cm, 4 cm, and 5 cm.
• Triangle B: Sides of length 6 cm, 8 cm, and 10 cm.

Check angles:

• All corresponding angles are equal.

Calculate scale factor:

• Side AB of Triangle A corresponds to side DE of Triangle B: 6 cm / 3 cm = 2
• Side BC of Triangle A corresponds to side EF of Triangle B: 8 cm / 4 cm = 2
• Side CA of Triangle A corresponds to side FD of Triangle B: 10 cm / 5 cm = 2

Since the scale factor is the same (2) for all sides, the triangles are similar.

Using Similarity in Problems

Similarity is a useful concept in geometry and can be used to solve various problems:

• Finding missing sides: If you know the scale factor and some side lengths, you can find missing sides.
• Calculating areas and volumes: The ratio of areas of similar shapes is the square of the scale factor, and the ratio of volumes is the cube of the scale factor.

Example Problem

Two rectangles are similar. The smaller rectangle has a length of 5 cm and a width of 3 cm. The larger rectangle has a length of 10 cm. What is the width of the larger rectangle?

Solution:

1. Scale factor: The length of the larger rectangle is double the length of the smaller rectangle (10 cm / 5 cm = 2).
2. Width: The width of the larger rectangle is also double the width of the smaller rectangle (3 cm * 2 = 6 cm).

Therefore, the width of the larger rectangle is 6 cm.