Answer:
6√2
Explanation:
You want one side of a square whose half-diagonal has length 6.
Right triangle
If X is the center point of the square where the diagonals cross at right angles, triangle EXG is an isosceles right triangle with leg length 6. The hypotenuse of an isosceles right triangle is √2 times the leg length, so ...
EG = 6√2
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Additional comment
If you like, you can use the Pythagorean theorem to prove that.
EG² + EX² +GX² = 6² +6² = 72
EG = √72 = √(36·2) = 6√2
A square is a rhombus, so the diagonals cross at right angles. A square is a rectangle, so the diagonals are the same length. A square is a parallelogram, so the diagonals bisect each other.