# The area of the triangle whose altitudes have lengths 36.4, 39, and 42 can be written as mn, where m and n are relatively prime positive integers. Find m + n.

m + n =7 + 507 = 514, assuming what is required is 4 times the area instead of the area.

## Expert Answers

We are given the altitudes of a triangle as 36.4, 39, and 42. We can find the area of a triangle if we know the three altitudes using this formula:

`A^(-1)=4sqrt(H(H-h_1^(-1))(H-h_2^(-1))(H-h_3^(-1)))`, where

`H=(h_1^(-1)+h_2^(-1)+h_(3)^(-1))/2` and `h_i` is the ith altitude.

Then, `H=(1/36.4+1/39+1/42)/2=1/26` .

So the inverse of the area is given as

`A^(-1)=4sqrt(1/26(1/26-1/36.4)(1/26-1/39)(1/26-1/42))`

`=4sqrt(4/50381604)=4(1/3549)`

Then the area can be expressed as `A=1/4(3549) ==> 4A=3549`

Note that the area is not a whole number and therefore cannot be expressed as the product of whole numbers. We can write 3549 = 7 x 507, where 7 and 507 are relatively prime.

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