Final answer:
When the area of a right-angled triangle is constant, using calculus and the Pythagorean theorem proves that the hypotenuse is least when the triangle is isosceles, making the statement true.
Step-by-step explanation:
To prove that a right-angled triangle with a constant area has the smallest hypotenuse when the triangle is isosceles, we need to use the properties of the triangle and calculus. We know the area of a right-angled triangle is ½ × base × height. Let's say the two legs are 'a' and 'b', and the hypotenuse is 'c'. The area is ½ × a × b. Because we're given that the area is constant, we can say that 'ab' is a constant value, let's call it 'k'. Hence, a = k/b. Using the Pythagorean theorem, we have a² + b² = c². Substituting 'a' with 'k/b', we have (k/b)² + b² = c². The hypotenuse 'c' is least when the derivative of 'c' with respect to 'b' is zero. Differentiating and setting the derivative equal to zero, we find that this occurs when 'a' equals 'b'. Thus, the hypotenuse 'c' is least when the triangle is isosceles. Therefore, the statement is true.