# Calculate height of triangle with little data and not trigonomery?

This ought to be a very simple question, but I seem to be completely stuck. I have a triangle, see image, and I'm supposed to calculate x, in this case it's height, without trigonometry. I believe the upper angle is meant to be 90 degrees.

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what are 18,4? Are they the lengths of the two segments? –  Dustan Levenstein Mar 31 '12 at 19:10
I can't imagine it would be something else, but it's actually not explicitly stated anywhere. –  user50849 Mar 31 '12 at 19:14
Hint: All the three triangles are similar. –  Salech Alhasov Mar 31 '12 at 19:18
Or look here: en.wikipedia.org/wiki/Right_triangle –  Salech Alhasov Mar 31 '12 at 19:24
Thanks for the input @SalechAlhasov. :) –  user50849 Mar 31 '12 at 19:43

But the area of the big triangle is also half of base ($18.4$) times height ($x$). So we have, $$\frac{1}{2}(12)\sqrt{(18.4)^2-12^2}=\frac{1}{2}(18.4)x.$$ The rest is calculator work.
Alternately, though this gets close to trigonometry, we can as before use the Pythagorean Theorem to find the third side of the big triangle, and use the fact that either of the smaller triangles is similar to the big one. You will get essentially the same equation as the one above, probably in the form $$\frac{x}{12}=\frac{\sqrt{(18.4)^2-12^2}}{18.4}.$$