Pythagoras' TheoremA self marking exercise on the application of Pythagoras' Theorem. 
Calculate the length of the third side of these right angled triangles. The diagrams are not to scale. Give your answer correct to 1 decimal place.
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Level 1  Finding the hypotenuse
Level 2  Finding a shorter side
Level 3  Mixed questions
Level 4  Pythagoras coordinates
Level 5  Mixed exercise
Level 6  More than one triangle
Level 7  Harder exercise
Exam Style questions requiring an application of Pythagoras' Theorem and trigonometric ratios to find angles and lengths in rightangled triangles.
Three Dimensions  Three dimensional Pythagoras and trigonometry questions
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See the National Curriculum page for links to related online activities and resources.
The area of the square on the hypotenuse of a right angled triangle is equal to the sum of the areas of the squares on the two shorter sides.
You may have learned the theorem using letters to stand for the lengths of the sides. The corners (vertices) of the rightangled triangle is labelled with capital (upper case) letters. The lengths of the sides opposite them are labelled with the corresponding small (lower case) letters.
Alternatively the sides of the rightangled triangle may me named using the capital letters of the two points they span.
As triangle can be labelled in many different ways it is probably best to remember the theorem by momorising the first diagram above.
To find the longest side (hypotenuse) of a rightangled triangle you square the two shorter sides, add together the results and then find the square root of this total.
To find a shorter side of a rightangled triangle you subtract the square of the other shorter side from the square of the hypotenuse and then find the square root of the answer.
AB^{2} = AC^{2}  BC^{2}
AB^{2} = 4.7^{2}  4.1^{2}
AB^{2} = 22.09  16.81
AB^{2} = 5.28
AB = √5.28
AB = 2.3m (to one decimal place)
The diagrams aren't always the same way round. They could be rotated by any angle.
The rightangled triangles could be long and thin or short and not so thin.
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QI,
Wednesday, June 12, 2019
"The Babylonians were using Pythagoras' Theorem over 1,000 years before Pythagoras was born."
Ann Roberts, London
Thursday, October 1, 2020
"Three D Pythagoras
Suppose you have a cuboid with length l, width w and height h.
Can you find the longest internal length d from one corner to the opposite corner of the box, in terms of l, w and h ?
Your answer is the 3Dpythagoras formula.
NOTE: Being able to apply the 2D Pythagoras formula to 3D shapes is still an essential skill, especially if you have a more complex 3D shape."