year 8 maths

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Introducing Pythagoras’ theorem
173
■ The hypotenuse
• It is the longest side of a right-angled triangle.
• It is opposite the right angle.
■ Pythagoras’ theorem
• The square of the hypotenuse is the sum of the squares
of the lengths of the other two shorter sides.
• a 2 + b 2 = c 2 or c 2 = a 2 + b 2
a
c
Hypotenuse
b
Key ideas
■ A Pythagorean triad (or triple) is a set of three integers that satisfy Pythagoras’ theorem.
Example 23
Checking Pythagorean triads
Decide if the following are Pythagorean triads.
a 6, 8, 10 b 4, 5, 9
SOLUTION
a a 2 + b 2 = 62 + 82
= 36 + 64
= 100 (=10 2 )
∴ 6, 8, 10 is a Pythagorean triad.
b a 2 + b 2 = 4 2 + 5 2
= 16 + 25
= 41
≠ 9 2
∴ 4, 5, 9 is not a Pythagorean triad.
EXPLANATION
Let a = 6, b = 8 and c = 10 and check that a 2 + b 2 = c 2 .
a 2 + b 2 = 41 and 9 2 = 81 so a 2 + b 2 ≠ c 2 .
Example 24
Deciding if a triangle has a right angle
Decide if this triangle has a right angle.
7 m
4 m
9 m
SOLUTION
a 2 + b 2 = 42 + 72
= 16 + 49
= 65 (≠ 9 2 = 81)
So the triangle does not have a right angle.
EXPLANATION
Check to see if a 2 + b 2 = c 2 . In this case
a 2 + b 2 = 65 and c 2 = 81, so the triangle is
not right angled.
Cambridge Maths NSW
Stage 4 Year 8 Second edition
ISBN 978-1-108-46627-1 © Palmer et al. 2018
Cambridge University Press
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