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The first step of problem solving is to understand what the problem is asking, that is where I am stuck.

One of the legs of a right triangle has length 4 cm. Express the length of the altitude perpendicular to the hypotenuse as a function of the length of the hypotenuse.

enter image description here

This is the picture I first came up with, since a is perpendicular to h. But the answer is $4\sqrt{h^2-16}/h$.

This means they do not really want $a$, as that would seem to be half $h$.

I need a fourth opinion here, I have asked others and no luck here.

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your picture is wrong $b=4$ is not necessaryly true. $a=h/2$ holds only for b=4. – sigmatau Jul 22 '13 at 22:26
Compare $a/4$ and $b/h$. – Etienne Jul 22 '13 at 22:28
up vote 2 down vote accepted

Note that $b=\sqrt{h^2-16}$. Also, by similar triangles (or using the sine of the angle at the bottom left corner), $$\frac{a}{4}=\frac{\sqrt{h^2-16}}{h}.$$

Remark: The reason you say that $a$ is half of $h$ is that your picture has $b$ almost the same length as the bottom side, so the whole picture looks like a square. But it need not be. The side $b$ could be a lot bigger or smaller than $4$.

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Yeah that clarifies things greatly, I realized that a more rectangle looking shape would make a non-perpendicular. – Leonardo Jul 22 '13 at 23:57

Their formula does yield $h$. The area of the triangle whose sides are $4$, $b$, and $h$ is $\frac12(ah)$. That area is also $\frac12(4b)=2b$. Thus, $\frac12(ah)=2b$, $ah=4b$, and $h=\frac{4b}a$. Now use the Pythagorean theorem to find $b$ in terms of $h$, and you have their answer.

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