Right integral triangles
Answer true or false on the following statement:
An integral right angle triangle can have its area as a perfect square number.
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1 solution
This is not correct. Which integers n , a correspond to a 5 , 1 2 , 1 3 right triangle?
(Also: Why can't the product of three consecutive (positive) integers be a perfect square?)
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For all positive integers n , there is no common factor of n and n 2 − 1 . Even if n be a perfect square, n 2 − 1 is never a perfect square. So n ( n 2 − 1 ) is never a perfect square. Regarding your first question, I have changed the expressions for the side lengths of the triangle.
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Agreed on your argument for the second question.
After the edit, this argument continues to need more detail. If x is rational, why can't x and x 2 − 1 be perfect squares? Note that x 2 − 1 can be a perfect square, e.g. x = 5 / 4 . You need more explanation.
The sides of an integral right triangle are of the form a 2 − b 2 , 2 a b , a 2 + b 2 , where a , b are positive integers. Area of such a triangle is a b ( a 2 − b 2 ) = b 4 x ( x 2 − 1 ) , where x = b a . Now either one or none of x and x 2 − 1 can be a square. Hence the area can never be a perfect square .