What is the greatest possible perimeter of a right-angled triangle with interger side lengths if one of the sides has length 12 ?

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Solution

$B y p y t h a g o r e a n t r i p l e t, t h e l e n g t h o f o t h e r t w o s i d e s c a n b e$
$2 m, m^{2} - 1 a n d m^{2} + 1.$
$L e t t h e l e n g t h o f t h e s h o r t e s t s i d e o f t h e r i g h t$
$∠ l e d △ l e b e 12 u n i t s .$
$⟹ 2 m = 12 ∴ m = 6 u n i t s .$
$P u t, m = 6 i n m^{2} - 1 & m^{2} + 1$
$⟹ m^{2} - 1 = 6^{2} - 1 = 35$
$m^{2} + 1 = 6^{2} + 1 = 37$
$∴ T h e l e n g t h o f t h e 3 s i d e s o f t h e r i g h t ∠ l e d △ l e a r e 12, 35 a n d 37.$
$∴ P e r i m e t e r o f △ l e = 12 + 35 + 37 = 84 u n i t s .$

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