# Right-angled triangles

Pythagoras' theorem states that for all right-angled triangles, 'The square on the 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.