9514 1404 393
Answer:
y = 27
Explanation:
The altitude of a right triangle creates two triangles that are each similar to each other and to the larger right triangle. This means corresponding sides are proportional.
If we write the proportion for the legs, we get ...
(long leg) / (short leg) = y/18 = 18/12
Multiplying by 18 gives us ...
y = 18(18/12)
y = 27
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Additional comment
The leg/leg proportion above gave rise to the relation ...
altitude² = (left hypotenuse segment)×(right hypotenuse segment)
That is, the altitude is the geometric mean of the two hypotenuse segments it touches. 18 = √(12y)
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There are two other "geometric mean" relationships in this triangle.
- The upper left side is the geometric mean of the left hypotenuse segment and the whole hypotenuse (the two segments it touches at the bottom).
- The upper right side is the geometric mean of the right hypotenuse segment and the whole hypotenuse (the two segments it touches at the bottom).
Each of these relationships is ultimately derived from the fact that all of the triangles are similar. You really only need to remember that these triangles are all similar and corresponding sides of similar triangles are proportional. (In some cases, it can be a bit of a shortcut if you remember the geometric mean relations.)