Pythagoras' theorem is a formula you can use to calculate the length of any of the sides on a right-angled triangle or the distance between two points.

Part of

Pythagoras' theorem states that for all right-angled triangles, **'The square on the** hypotenuse **is equal to the sum of the squares on the other two sides'**. The hypotenuse is the longest side and it's always opposite the right angle.

In this triangle \(a^2 = b^2 + c^2\) and angle \(A\) is a right angle.

Pythagoras' theorem only works for right-angled triangles, so you can use it to test whether a triangle has a right angle or not.

In the triangle above, if \({a}^{2}~\textless~{b}^{2}+{c}^{2}\) the angle \(A\) is acute.

In the triangle above, if \({a}^{2}~\textgreater~{b}^{2}+{c}^{2}\) the angle \(A\) is obtuse.

- Question
Which of the following triangles is right-angled?

**a)****b)****c)****a)**\[8^2 = 64\]

\[5^2 + 6^2 = 25 + 36 = 61\]

\(8^2\) is greater than \(5^2 + 6^2\),

so angle \(A\) must be

**obtuse**.**b)**\[13^2 = 169\]

\[5^2 + 12^2 = 25 + 144 = 169\]

\(13^2\) is equal to \(5^2 + 12^2\)

so angle \(P\) must be a

**right angle**.**c)**\[10^2 = 100\]

\[5^2 + 9^2 = 25 + 81 = 106\]

\(10^2\) is less than \(5^2 + 9^2\)

so angle \(X\) must be

**acute**.So, it is triangle

**b**which is right-angled.