1. Problem statement: In the right triangle with vertices G (top-left), F (bottom-left), and H (bottom-right), the side GF is the opposite side with length 6 and the side FH is the adjacent side with length 18, and the hypotenuse GH is unknown. 2. Formula and rules: Use the Pythagorean theorem for right triangles: the square of the hypotenuse equals the sum of the squares of the other two sides. In symbols: $$hyp^2 = opp^2 + adj^2$$ Important rule: side lengths are nonnegative and the hypotenuse is the positive square root. 3. Calculation: Compute the squares of the legs and sum them. Compute the square of the opposite side: $$6^2 = 36$$ Compute the square of the adjacent side: $$18^2 = 324$$ Sum the squares: $$36 + 324 = 360$$ Apply the Pythagorean theorem: $$hyp = \sqrt{360}$$ Simplify the radical by factoring: $$\sqrt{360} = \sqrt{36\cdot 10} = 6\sqrt{10}$$ Decimal approximation: $$6\sqrt{10} \approx 18.97366596$$ 4. Final answer: The exact length of the hypotenuse GH is $$6\sqrt{10}$$. The approximate length is $$18.97366596$$.