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2 Shapes and symmetry

2.1 Geometric shapes – triangles

This section deals with the simplest geometric shapes and their symmetries. All of the shapes are two-dimensional – hence they can be drawn accurately on paper.

Simple geometric shapes are studied in mathematics partly because they are used in thousands of practical applications. For instance, triangles occur in bridges, pylons and, more mundanely, in folding chairs; rectangles occur in windows, cinema screens and sheets of paper; while circles are an essential part of wheels, gears and plates.

By definition, triangles are shapes with three straight sides. However, there are various types of triangle:

An equilateral triangle is a triangle with all three sides of equal length. The three angles are also all equal.

An isosceles triangle is a triangle with two sides of equal length. The two angles opposite the equal sides are also equal to one another.

A right-angled triangle is a triangle with one angle that is a right angle.

A scalene triangle is a triangle with all the sides of different lengths. The angles are also all different.

It is a general convention that equal sides are marked by drawing a short line, /, through them, and a right angle is marked by a square between the arms of the angle. If sides and angles are not marked, do not assume that they are equal, just because they look equal!


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