Answer:
- x = 10.68
- y = 12.25
- z = 21.79
Explanation:
You want the side lengths in a right triangle geometry when the altitude to the hypotenuse of length 25 cuts off a segment of length 6.
Geometric mean relations
All of the right triangles in this configuration are similar. The proportional relationships between the sides give rise to three geometric mean relations.
In effect each of the segments x, y, z is the geometric mean of the two segments of the hypotenuse it touches.
x = √(6·(25-6)) = √114 ≈ 10.68
y = √(6·25) = 5√6 ≈ 12.25
z = √((25 -6)·25) = 5√19 ≈ 21.79
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Additional comment
To see how these arise, consider the ratio of long side to hypotenuse:
z/25 = 19/z
z² = 25·19
z = √(25·19) . . . . . the geometric mean of 25 and 19
The "geometric mean" of n numbers is the n-th root of their product. For two numbers, it is the square root of their product.