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 Three Altitudes (Posted on 2007-01-15)
If the lengths of the altitudes of a triangle are 4, 5, and 6, what is the area of the triangle?

Can you generalize?

 See The Solution Submitted by Dennis Rating: 3.0000 (1 votes)

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 Solution | Comment 1 of 3
`Let a, b, and c be the lengths of the sidesof the triangle and h_a, h_b, and h_c thecorresponding altitudes. Then, `
`  Area = a*h_a/2 = b*h_b/2 = c*h_c/2      (1)`
`If s is the semiperimeter of the triangle,then by Heron's formula we have`
`  Area = sqrt(s(s-a)(s-b)(s-c))           (2)`
`Combining (1) and (2) with a lot of algebra,we get`
`  Area = x*y*z/(4*sqrt(w(w-x)(w-y)(w-z)), `
`  where`
`    x = h_a*h_b    y = h_b*h_c    z = h_c*h_a    w = (x+y+z)/2`
`For our problem,`
`    x = 4*5 = 20    y = 5*6 = 30    z = 6*4 = 24    w = (20+30+24)/2 = 37`
`    and `
`    Area = 3600/sqrt(57239)`
`         ~= 15.04723`
` `

 Posted by Bractals on 2007-01-15 17:43:05

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