Final answer:
The hypotenuse of a right triangle circumscribing a circle is the diameter of that circle. This follows from the Thales' theorem and the properties of a circumcircle around a right triangle.
Step-by-step explanation:
When constructing a right triangle and circumscribing a circle about it, the relationship that the hypotenuse has with the circle is that the hypotenuse is the diameter of the circle. This is because according to the properties of a circumscribed circle (or circumcircle) around a right triangle, the right angle is subtended by a semicircle or half circle. Therefore, the side opposite the right angle, which is the hypotenuse, spans the diameter of the circumcircle. This relationship is a consequence of the Thales' theorem, which states that any triangle inscribed in a circle such that one of the triangle's sides is a diameter of the circle will be a right triangle.
To clarify this with examples of the Pythagorean theorem, the side lengths a and b would be considered legs, and the side c which is the hypotenuse, is equal to the square root of the sum of the squares of the other two sides, or c = √(a² + b²). If you consider a triangle with these side lengths inscribed in a circle, the hypotenuse c would span from one end of the circle to the other, making it the diameter.