Mathematics (Core)Exam code: 0580

Pythagoras

Pythagoras and Trigonometry

PractiseFlashcards

Pythagoras' Theorem

The relationship

  • Pythagoras' theorem links the three sides of a right-angled triangle, and it works in no other triangle
  • The longest side is the hypotenuse, and it is always the one opposite the right angle
  • The theorem states a² + b² = c², in which c stands for the hypotenuse while a and b stand for the other two sides
  • Whichever of those two you call a and whichever you call b makes no difference to the result
A right-angled triangle with the right angle marked by a small square, the two shorter sides labelled a along the base and b up the side, and the sloping side opposite the right angle labelled c as the hypotenuse
Source: Pythagoras theorem by Save My Exams
  • Papers do not use the words "Pythagoras" or "hypotenuse" in the question: they simply give a right-angled triangle with two sides marked

Using it both ways

  • To find the hypotenuse, square both shorter sides, add, then take the square root
  • To find a shorter side, square the hypotenuse and the known shorter side, subtract, then take the square root
  • Adding when you should subtract is the standard error, and the check is simple: the hypotenuse must come out as the longest side
Triangle ABC with B above the base AC and a perpendicular BD dropped onto it: AB is 12 cm, AD is 9 cm and the whole base AC is 22 cm, so BD comes from one right-angled triangle and the unknown side x from a second
Source: Pythagoras theorem by Save My Exams
  • Where an answer does not work out exactly, round it as the question instructs, and to 3 significant figures when it does not say
  • Cut a bigger figure into right-angled pieces and the theorem applies again, so dropping a perpendicular inside an isosceles triangle or a trapezium opens it up

Exam tip

The squares line earns a mark

Pythagoras was set in 10 of the 22 papers. Every scheme gives a method mark for the line with both squares in place, such as 6² + 9², or 11² − 6² for a shorter side, even if the root or rounding then slips. Find any side that is not printed first: an isosceles triangle 10 cm wide and 12 cm tall halves into triangles 5 cm wide, so the line is 12² + 5², not 12² + 10².

Worked example

Both directions in one triangle

A right-angled triangle has shorter sides 9 cm and 12 cm. Find the hypotenuse. A second right-angled triangle has hypotenuse 25 cm and one shorter side 7 cm. Find the other shorter side.

Two separate right-angled triangles. The first has its right angle marked, a vertical side labelled 9 cm and a horizontal side labelled 12 cm, with the sloping side unlabelled. The second has its right angle marked, a short vertical side labelled 7 cm and a long sloping side labelled 25 cm, with the horizontal side unlabelled.

Solution:

  • For the first, the unknown is the hypotenuse, so add
  • c² = 9² + 12² = 81 + 144 = 225
  • c = √225 = 15 cm
  • For the second, the unknown is a shorter side, so subtract
  • a² = 25² − 7² = 625 − 49 = 576
  • a = √576 = 24 cm
  • Check both: 15 is longer than 9 and 12, and 24 is shorter than 25

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