Right Triangle Chain

Geometry Level 2

All of the triangles are right-angled with integer sides. If $AB = 3$ , what is the length of $AE$ ?

65 85 75 13

1 solution

Maria Kozlowska
Mar 5, 2016

The only Pythagorean triples possible are:

$3^2+4^2=5^2$

$5^2+12^2=13^2$

$13^2+84^2=85^2$

This can be shown using Euclid's formula. Euclid's formula is a fundamental formula for generating Pythagorean triples given an arbitrary pair of positive integers m and n with $m > n$ . The formula states that the integers $a = m^2 - n^2 , b = 2mn , c = m^2 + n^2$

form a Pythagorean triple.

$3=m^2-n^2=(m-n)(m+n) \Rightarrow m-n=1, 2n+1=3, m=2,n=1, c=5$

$5=m^2-n^2=(m-n)(m+n) \Rightarrow m-n=1, 2n+1=5, m=3,n=2, c=13$

$13=m^2-n^2=(m-n)(m+n) \Rightarrow m-n=1, 2n+1=13, m=7,n=6, c=\boxed{85}$

Show that this is the only possible integral solution.

Kushagra Sahni - 5 years, 3 months ago

As mentioned, "The only possible Pythagorean triples". We are somewhat lucky that these legs produced unique lengths, which forced the next value.

In fact, the interesting question would be with regards to the 4th triangle. What would be the minimum hypotenuse then? Perhaps you can pose that problem :)

Calvin Lin Staff - 5 years, 3 months ago

Right Triangle Chain

1 solution

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