Find the lengths of the sides of the right triangle?

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


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.

Yes, two vertical lines can be complementary