A right triangle has legs a and b, and hypotenuse c.
The length of the median to the hypotenuse is (a^{3}b + b^{3}a)^{1/3}.
Find the length of the altitude to the hypotenuse
In our right triangle: a=angle between leg a and hypotenuse c; h=altitude
sin(a)=b/c=h/a => h=ba/c
median lenght=c/2 as is evident if designing the whole rectangle
So:
c/2=(a^{3}b + b^{3}a)^{1/3}=ab(a^{2} + b^{2})^{1/3} =(abc^{2} )^{1/3}.
As ba=hc
c/2=(hc^{3})^{1/3}=c*h^{1/3}. Then: 1/2=h^{1/3}.
h=1/8
h is invariant for all triangles that verify that lenght of median

Posted by armando
on 20190313 10:46:14 