Question

The altitude to the hypotenuse of a right triangle divides the hypotenuse into 15 inches and 30 inch segments. Explain or show how to find the exact length of the altitude including the correct value. Then explain or show how to find the length of each leg including the correct values.

Answer 1

9514 1404 393

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.