What you need to remember about this geometry is that all of the triangles are similar. As with any similar triangles, that means ratios of corresponding sides are proportional. Here, we can write the ratios of the long leg to the short leg and set them equal to find x.

x/60 = 60/25

Multiply by 60 to find x:

x = (60·60)/25

x = 144

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<em>Comment on this geometry</em>

You may have noticed that the above equation can be written in the form ...

60 = √(25x)

That is, the altitude from the hypotenuse (60) is equal to the geometric mean of the lengths into which it divides the hypotenuse (25 and x).

This same sort of "geometric mean" relation holds for other parts of this geometry, as well. The short leg of the largest triangle (the hypotenuse of the one with legs 25 and 60) is the geometric mean of the short hypotenuse segment (25) and the total hypotenuse (25+x).

And, the long leg of the large triangle (the hypotenuse of the one with legs 60 and x) is the geometric mean of the long hypotenuse segment (x) and the total hypotenuse (25+x).

While it can be a shortcut in some problems to remember these geometric mean relationships, you can always come up with what you need by simply remembering that the triangles are all similar.

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3 years ago