The hypotenuse of a right-angled triangle is 12 cm. What values can the area of this triangle take?

Let a right-angled triangle ABC have the following parameters: hypotenuse AB = 12 cm, legs AC = b and BC = a, area S (Δ ABC) = S, and S = a ∙ b / 2. Then b = 2 ∙ S / a. By the Pythagorean theorem AB² = AC² + BC², we get 12² = b² + a² or 12² = (2 ∙ S / a) ² + a². Simplifying, we got a biquadratic equation for a: (a²) ² – 12² a² + 4 ∙ S² = 0; estimate its discriminant; D = (12²) ² – 4 ∙ 4 ∙ S²; D ≥ 0; (12²) ² – 4 ∙ 4 ∙ S² ≥ 0; (12²) ² ≥ 4 ∙ 4 ∙ S²; 1296 ≥ S²; 36 ≥ S.
Answer: the area of the triangle does not exceed 36 cm².