What is Pythagoras' Theorem and how is it used in exam questions?

Pythagoras' Theorem is an equation used to give us more information about right angled triangles. The theorem itself looks like this: a²+b²=c² where c = the hypotenuse (the longest side of the triangle, always opposite the right angle) and a and b are the other two sides of the triangle, either side of the right angle. If we don't know the length of one of the sides of a triangle, we can use Pythagoras' Theorem to figure it out! For example, if we know that a = 3cm and b = 4cm, we can find c accordingly: a²+b²=c² , 3²+4²=c² , 9 + 16 = c² , 25 = c² , c = 5 We can also use the equation when we don't know the value of a or b. For example: a²+b²=c² , a²+5²=8² , a²+25=64 , a²=64-25 , a²=39 , a= the square root of 39. The theorem is often used in exam questions to either: discover an unknown length of the side of a right angled triangle; to prove whether a triangle is a right angled triangle or not; to find the length of a line on a graph; and to solve geometry problems regarding both 2D and 3D shapes.