Pythagoras' theorem states that in any right-angled triangle the square of the longest side (called the HYPOTENUSE) is equal to the sum of the squares of the other two shorter sides.If the two shortest sides are a and b and the hypotenuse is c then Pythagoras' theorem states: c² = a² + b².Pythagoras' theorem can be used to: 1) Identify if a triangle is right-angled, e.g. If a triangle has sides of 5cm, 8cm and 6cm is it right-angled? In this case 8cm is the hypotenuse therefore if it is right-angled then 8² should be equal to 5² + 6². However 5²+ 6²= 25 + 36 = 61, whereas 8² = 64, therefore it is NOT right-angled. 2) Calculate the length of any side in a right-angled triangle, e.g. If a right-angled triangle has a hypotenuse of 5cm and one of the other sides is 2cm calculate the length of the other side (X). 5² = 2² + X², rearrange the equation to make X the subject, X² = 5² - 2². X² = 25 - 4, X = √21 = 4.58 cm (to 2dp). 3) Calculate the difference between 2 points (the length of a line segment), e.g. If point A has the coordinates (2,10) and B has the coordinates (6, 4) calculate the distance between point A and B. In this case, AB will be the hypotenuse and the length of each side will be the largest minus the smallest (6-2, 10-4) = (4, 6). Therefore AB² = 4² + 6², AB²= 16 + 36, AB = √52, AB = 7.21 (to 2dp). 4) Solve three-dimensional problems where there are right-angled triangles involved (HIGHER), e.g. If a cuboid ABCDEFGH has equal sides of 4cm, what is the length of A to G? Draw the right-angled triangle involved AG, and you should get AG² = 4² + X². To figure out X you can draw another right angled triangle where X² = 4² + 4², therefore X = √(16 +16) = 5.66 cm. Plug the value of X back into your initial equation, AG² = 4² + 5.66², AG = √(16 +32.04) = 6.93 cm.

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