Find the lengths of the sides of the right triangle?

• sqdancefan: You can guess from your knowledge of Pythagorean triples that the triangle is a 5-12-13 right triangle. Since one side is 7 less than the other, this guess is confirmed.
  
  The lengths of the sides are:
    2x = 12
    2x -7 = 5
  
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  You can use the Pythagorean theorem to write the relationship between the sides.
    (2x)² +(2x -7)² = 13²
    4x² +4x² -28x +49 = 169 . . . . . eliminate parentheses
    8x² -28x -120 = 0 . . . . . . . . . . . put in standard form
    4(2x² -7x -30) = 0 . . . . . . . . . . . factor out 4
    (x -6)(2x +5) = 0 . . . . . . . . . . . .. divide by 4 and factor further
    x = 6, -5/2 . . . . . . . . . . . . . . . . .. only the positive solution (x=6) is useful
  Then 2x = 12; 2x-7 = 5.