Understanding Engineering Mathematics

We can see that this works by checking it for a few simple triangles – the 3 : 4 : 5 being the
classic example. But this is not aproof– it is merelyinductionfrom a few cases, notdeduc-
tionfrom fundamental axioms. There are numerous proofs of Pythagoras’ theorem – we will
give one which has the sort of common sense appeal an engineer might enjoy.
Take four identical copies of any right-angled triangle, sidesa,b,cand arrange them
as shown in Figure 5.19, to form a square of sidea. While you are at it, just check
that all yourvisualarguments can be expressed solely in purely symbolic or geometrical
terms – imagine, for example, that you could not see or draw the diagram, and that you
had to describe and justify the whole thing to a friend.
Now from trianglesAEDandDHC,DH=candDE=b,soHE, the side of the
small internal square isb−c. The same argument applies to all pairs of adjacent triangles.

AB

DC

a H

E

a

c

a

a

b

Figure 5.19Proof of Pythagoras’ theorem.

( The total area of the big square is four times the area of the triangle
‘half base times height’=^12 bc

)

, plus the area of the small square. So

a^2 = 4 ×^12 bc+(b−c)^2

which with a bit of algebra (42
➤
) simplifies to

a^2 =b^2 +c^2

Solution to review question 5.1.8

Of the two given sides, the longest, 6, must be the hypotenuse. If the

shortest side isxthen we have

x^2 + 52 = 62

so

x=

√

62 − 52 =

√

11