Cambridge International Mathematics

The theorem of Pythagoras (Chapter 8) 171

There are over 400 different proofs of Pythagoras’ theorem. Here is one of them:

On a square we draw 4 identical (congruent) right angled triangles,

as illustrated. A smaller square is formed in the centre.

Suppose the legs are of lengthaandband the hypotenuse has

lengthc.

The total area of the large square

=4£area of one triangle+area of smaller square,

) (a+b)^2 =4(^12 ab)+c^2

) a^2 +2ab+b^2 =2ab+c^2

) a^2 +b^2 =c^2

Example 1 Self Tutor

Find the length of the hypotenuse in:

If x^2 =k, then

x=§

p

k, but

we reject ¡

p

k

as lengths must

The hypotenuse is opposite the right angle and has lengthxcm. be positive.

) x^2 =3^2 +2^2

) x^2 =9+4

) x^2 =13

) x=

p

13 fas x> 0 g

) the hypotenuse is about 3 : 61 cm long.

Example 2 Self Tutor

Find the length of the third side of this triangle:

The hypotenuse has length 6 cm.

) x^2 +5^2 =6^2 fPythagorasg

) x^2 +25=36

) x^2 =11

) x=

p

11 fas x> 0 g

) the third side is about 3 : 32 cm long.

b

ab

a

a

a

c

c

c

c

b

b

3cm

2cm

xcm

6cm

5cm

xcm

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