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Answer:
CX = 15√2 inches
BC = 15√3 inches
AC = 15√6 inches
Step-by-step explanation:
In this geometry, all of the triangles are similar:
ΔABC ~ ΔACX ~ ΔCBX
Corresponding segments are proportional in similar triangles, so we have ...
AX/CX = CX/BX = (long leg)/(short leg)
Filling in the numbers, we get
30/CX = CX/15
CX² = 15×30
CX = 15√2 . . . . . exact length of the altitude (inches)
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Similarly, we can form proportions with the legs:
AB/BC = CB/BX
BC² = (BX)(AB) = (45)(15)
BC = 15√3 . . . inches
and
AC/AB = AX/AC
AC² = (AX)(AB) = (30)(45)
AC = 15√6 . . . inches
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Additional comment
You may notice that each of the segments we're interested in is the root of a product involving segments of the hypotenuse. This "root of a product" is called the geometric mean. Here, the three geometric mean relations are ...
altitude = geometric mean of hypotenuse segments
short side = geometric mean of short segment and whole hypotenuse
long side = geometric mean of long segment and whole hypotenuse
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Strictly speaking the geometric mean is the n-th root of the product of n items. Here, there are only 2 items, so it is the square root of their product.
