Prove that there is no triangle whose altitudes are of length 4, 7, and 10 units.

The area found by base*height/2 should be the same for each of 3 sides, so given sides a, b, c

2a=7b/2=5c

Side by the triangle inequality b+c > a

solving for b in terms of a gives b = 4a/7
solving for c in terms of a gives c = 2a/5

b + c = 4a/7 + 2a/5 = 34a/35

Which is not greater than a so the triangle cannot exist.

Posted by Jer on 2013-10-04 09:07:31