Describe types of triangles and Pythagoras' theorem

Types of Triangles

A triangle is a shape with three sides and three angles. Triangles can be grouped in two ways: by their sides and by their angles.

Types of Triangles by Their Sides

Equilateral triangle has three equal sides and three equal angles (each 60°).

Isosceles triangle has two equal sides. The angles opposite the equal sides are also equal.

Scalene triangle has three different sides and three different angles.

Types of Triangles by Their Angles

Acute triangle has all three angles less than 90°.

Right-angled triangle has one angle exactly 90°. The two sides that form the right angle are called the base and the height. The side opposite the right angle is called the hypotenuse — it is the longest side.

Obtuse triangle has one angle greater than 90°.

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Pythagoras' Theorem

The Pythagoras' theorem applies only to right-angled triangles. It states:

The square of the hypotenuse equals the sum of the squares of the other two sides.

In symbols:

$a^2 + b^2 = c^2$

Where:

• $a$ = length of the base
• $b$ = length of the height
• $c$ = length of the hypotenuse

Finding the Hypotenuse

When you know the base ($a$) and height ($b$), find the hypotenuse ($c$) using:

$c = \sqrt{a^2 + b^2}$

Example: Find the hypotenuse of a right-angled triangle with base 8 cm and height 15 cm.

Solution: $c^2 = a^2 + b^2$ $c^2 = 8^2 + 15^2$ $c^2 = 64 + 225$ $c^2 = 289$ $c = \sqrt{289} = 17 \text{ cm}$

Finding the Base or Height

When you know the hypotenuse ($c$) and one side ($a$ or $b$), find the missing side using:

$a = \sqrt{c^2 - b^2}$ $b = \sqrt{c^2 - a^2}$

Example: A ladder 13 m long leans against a wall. The foot of the ladder is 5 m from the wall. How high does the ladder reach?

Solution:

• Base ($a$) = 5 m
• Hypotenuse ($c$) = 13 m
• Height ($b$) = ?

$b = \sqrt{c^2 - a^2}$ $b = \sqrt{13^2 - 5^2}$ $b = \sqrt{169 - 25}$ $b = \sqrt{144} = 12 \text{ m}$

The ladder reaches 12 metres up the wall.

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Real-life application

Masons in Tanzania use the Pythagoras' theorem when building houses. For example, to make sure two walls meet at a right angle, they measure three sides of 3 metres, 4 metres, and 5 metres — if the 5-metre diagonal fits exactly, the corner is perfectly right-angled. This method is called the "3-4-5 rule" and is commonly used at construction sites across Tanzania.