Question 909429

A triangle has side lengths measuring {{{a=30}}}, {{{b=40}}}, and {{{c=50}}} units is an right angle triangle.

In a right triangle the three altitudes are: {{{h[a]}}}, {{{h[b], and {{{h[c]}}} (the first two of which equal the leg lengths {{{a}}} and {{{b}}} respectively) and they are related according to


{{{1/h[a]^2+1/h[b]^2=1/ h[c]^2}}}

plug in {{{h[a]=30}}} and  {{{h[b]=40}}}

and find {{{h[c]}}}

{{{1/30^2+1/40^2=1/ h[c]^2}}}

{{{1/900+1/1600=1/ h[c]^2}}}

{{{1600/1440000+900/1440000=1/ h[c]^2}}}

{{{(1600+900)/1440000=1/ h[c]^2}}}

{{{25cross(0)cross(0)/14400cross(0)cross(0)=1/ h[c]^2}}}


{{{25/14400=1/ h[c]^2}}}

{{{cross(25)1/cross(14400576)=1/ h[c]^2}}}


{{{1/576=1/ h[c]^2}}}

{{{h[c]^2=576}}}

{{{h[c]=sqrt(576)}}}

{{{h[c]=24}}} ...so, this is the length of shortest altitude