MathLabs

Problem 4

Construct a right triangle with a given hypotenuse cc such that the median drawn to the hypotenuse is the geometric mean of the two legs.
Step 2 of 4: Determine the altitude to the hypotenuse
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[ABC]=ab2=ch2⟹h=c4[ABC]=\frac{ab}{2}=\frac{ch}{2}\quad\Longrightarrow\quad h=\frac{c}{4}
Detailed analysis

If hh is the altitude to the hypotenuse, the area is both ab/2ab/2 and ch/2ch/2. Using ab=c2/4ab=c^2/4 gives ch/2=c2/8ch/2=c^2/8, so h=c/4h=c/4.