000RM.dvi

214 Pythagorean triples

6.5 Dissecting a rectangle into Pythagorean triangles

How many matches (of equal lengths) are required to make up the fol-
lowing figure?

a

b−y

y

x a−x

b v w

u

B

D C

A

Q

P

This is Problem 2237 of theJournal of Recreational Mathematics,
which asks for the
(i) the smallest such rectangle,
(ii) the smallest such rectangle withAP=AQ,
(iii) the smallest such rectangle withAP QPythagorean,
(iv) the smallest square.
In (iii), the three right trianglesQP C,AQDandAP Qare similar.
If they are similar to the Pythagorean trianglea : b : c, the ratios of
similarity are alsoa : b : c. If we putCP = a^2 , then the lengths
of the various segments are as shown below. Note thatABPnow is a
Pythagorean triangle with parametersbanda. With(a, b, c)=(3, 4 ,5),
we obtain the smallest rectangle satisfying (iv). This also answers (i)
since it is clearly also the smallest rectangle (with any restrictions).

2 ab

a^2

b^2 −a^2

ab ab

b^2

ac

bc

c^2 =a

(^2) +b^2
B
D C
A
Q
P
(ii) is also tractable without the help of a computer. Here, we want
two Pythagorean triangles with the same hypotenuse. It is well known
that the smallest such hypotenuse is 65. Indeed, 652 =63^2 +16^2 =
562 +33^2. From this it is easy to complete the data.
The answer to (iv) is given in the Appendix.