13.1 The Pythagoras’ Theorem - Mathematics

13.1 The Pythagoras’ Theorem

13.1.1 The Pythagoras’ Theorem

1. In a right-angled triangle, the hypotenuse is the longest side of the triangle.

2. Pythagoras’ Theorem:

In a right-angled triangle, the square

of the hypotenuse is equal to the sum

of the squares of the other two sides.

Example 1:

Solution:

\begin{matrix} x^{2} = 5^{2} + 12^{2} = 25 + 144 x = \sqrt{169} = 13 \end{matrix}

$\begin{array}{l} x^{2} = 5^{2} + 12^{2} \\ = 25 + 144 \\ x = \sqrt{169} \\ = 13 \end{array}$

Example 2:

Solution:

\begin{matrix} x^{2} = 15^{2} - 9^{2} = 225 - 81 x = \sqrt{144} = 12 \end{matrix}

$\begin{array}{l} x^{2} = 15^{2} - 9^{2} \\ = 225 - 81 \\ x = \sqrt{144} \\ = 12 \end{array}$

3. Pythagorean triples are three whole numbers that form the sides of a right-angled triangle.

Example:

(a) 3, 4, 5
(b) 6, 8, 10

(e) 9, 12, 15

13.1.2 The Converse of the Pythagoras’ Theorem

In a triangle, if the sum of the squares of the two sides

is equal to the square of the longest side, then the angle

opposite the longest side is a right angle.