Final answer:
The geometric mean theorem in a right triangle involves the altitude and segments of the hypotenuse, where the altitude is the geometric mean of the two segments it creates on the hypotenuse. Therefore, none of the provided proportions (a, b, c, d) are correct representations of the geometric mean theorem.
Step-by-step explanation:
The question pertains to the geometric mean theorem as it relates to right triangles, and specifically to the proportions involving the altitude (h), the legs (b and c), and the hypotenuse. The geometric mean theorem states that the altitude (h) from the right angle to the hypotenuse of a right triangle divides the hypotenuse into two segments, d and e, such that the altitude is the geometric mean of the segments: h2 = d × e. Additionally, each leg of the triangle is the geometric mean between the hypotenuse and the projection of that leg onto the hypotenuse. This means that for a right triangle, the proportion will look like b2 = h × d and c2 = h × e. Therefore, none of the provided options (a, b, c, d) correctly represent the geometric mean theorem.