What is Pythagoras' theorem and what can it be used to figure out?
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.
