Proof of Pythagoras theorem

The Pythagoras Theorem

Pythagoras' Theorem states:

In a right-angled triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse (the longest side).

This can be written as:

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

Where:

• a and b are the lengths of the shorter sides
• c is the length of the hypotenuse

Proof using geometry

Consider a large square formed with side length $(a + b)$, and inside it, place four identical right-angled triangles, each with sides $a$, $b$, and hypotenuse $c$. See the figure below:

Step 1: Area of the large square

Each side of the large square is $(a + b)$, so its area is:

$\text{Area}_{\text{large square}} = (a + b)^2$

Step 2: Area of the inner pieces

Inside the large square:

• There is one smaller (tilted) square formed by the hypotenuses of the triangles. Its area is:

$\text{Area}_{\text{inner square}} = c^2$

• There are 4 right-angled triangles. The area of each triangle is:

$\text{Area}_{\text{triangle}} = \frac{1}{2}ab$

• Total area of the 4 triangles is:

$4 \times \frac{1}{2}ab = 2ab$

Step 3: Total area in terms of parts

The total area of the large square equals the area of the inner square plus the area of the 4 triangles:

$(a + b)^2 = c^2 + 2ab$

Step 4: Expand the left side

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

Step 5: Subtract $2ab$ from both sides

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

Therefore, proved!

Note: Pythagoras' Theorem is useful in solving many problems involving right-angled triangles, especially when the lengths of two sides are known and the third is unknown.

Examples of Pythagoras theorem

Example 1: Finding the hypotenuse

A right-angled triangle has sides of length 3 cm and 4 cm. Find the length of the hypotenuse. Let: $a = 3 \text{ cm}$ $b = 4 \text{ cm}$ $c = \text{hypotenuse}$ Using Pythagoras' Theorem: $c^2 = a^2 + b^2 = 3^2 + 4^2 = 9 + 16 = 25$ $c = \sqrt{25} = 5 \text{ cm}$ Answer: The hypotenuse is 5 cm.

Example 2: Finding a side

In a right-angled triangle, the hypotenuse is 13 cm and one side is 5 cm. Find the other side. Let: $c = 13 \text{ cm}$ $a = 5 \text{ cm}$ $b = ?$ Using Pythagoras' Theorem: $c^2 = a^2 + b^2 \Rightarrow 13^2 = 5^2 + b^2 \Rightarrow 169 = 25 + b^2$ $b^2 = 169 - 25 = 144 \Rightarrow b = \sqrt{144} = 12 \text{ cm}$ Answer: The other side is 12 cm.

Example 3: Check if a triangle is right-angled

A triangle has sides of 6 cm, 8 cm, and 10 cm. Is it a right-angled triangle? Let: $a = 6 \text{ cm}, \quad b = 8 \text{ cm}, \quad c = 10 \text{ cm}$ Check if $a^2 + b^2 = c^2$: $6^2 + 8^2 = 36 + 64 = 100$, and $c^2 = 10^2 = 100$ Answer: Yes, it is a right-angled triangle.