Digit Reversal Triangle

Geometry Level 4

A right-angled triangle has integer side lengths less than 100. Reversing the digits in the hypotenuse length gives us the length of one of the other two sides. What is the length of the hypotenuse?

Source: 1995 Grade 12 Waterloo Math Contest - Question 8


The answer is 65.

Matt O by Matt O , 41, Canada

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1 solution

A 2 + ( 10 C 1 + C 2 ) 2 = ( 10 C 2 + C 1 ) 2 . H y > L e g ∴ C 2 > C 1 . A n d A 2 = 99 ( C 2 2 − C 1 2 ) . S i n c e a l l a r e i n t e g e r s , A 2 = 3 ∗ 3 ∗ 11 ∗ 11 = 3 3 2 . ⟹ ( C 2 2 − C 1 2 ) = 11. ∴ ( C 2 − C 1 ) ∗ ( C 2 + C 1 ) = 1 ∗ 11. ⟹ C 2 − C 1 = 1 , C 2 + C 1 = 11. S o H y . = 65. L o o k i n g u p i n P y t h a g o r a s T r i p l e l i s t , w i l l e a s i l y g i v e t h e s a m e a n s w e r . A^2+(10C_1+C_2)^2=(10C_2+C_1)^2.~~Hy>Leg~~\therefore~C_2>C_1.\\ And~A^2=99(C_2^2-C_1^2).\\ Since~all~are~integers,~~A^2=3*3*11*11=33^2.\\ \implies~(C_2^2-C_1^2)=11.~~~\therefore~(C_2-C_1)*(C_2+C_1)=1*11.\\ \implies~~C_2-C_1=1,~~~C_2+C_1=11.\\ So~Hy.=\Large~~\color{#D61F06}{65}. ~~\\ Looking~up~in~Pythagoras~ Triple~list,~~will~easily~give~the~same~answer.

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