   SEARCH HOME Math Central Quandaries & Queries  Question from grace, a student: Two of the altitudes of a scalene triangle ABC have length 4 and 12. If the length of the third altitude is also an integer, what is the biggest that it can be? Justify all of your conclusions. Grace,

You must know two basic facts about triangles to solve this problem:

1. THE PRODUCT OF THE LENGTHS OF A SIDE AND THE ALTITUDE TO THAT SIDE EQUALS TWICE THE AREA. So if denote twice the area by K and let a, b, and c be the sides corresponding to the altitudes of length 4, 12, and h, we get the formulas,

a = K/4, b = K/12, c = K/h

2. A TRIANGLE EXISTS WITH SIDE LENGTHS a, b, AND c IF AND ONLY IF THEY SATISFY THE THREE TRIANGLE INEQUALITIES:

a < b + c, b < c + a, and c < a + b.

At this point somebody who knows enough algebra can deduce the largest possible value of h immediately. If you are not confident with algebra, you can test the integer values of h starting at h = 1 (and skipping h = 4 and h = 12 to avoid isosceles triangles) to discover the largest value of h for which a, b, and c satisfy the triangle inequalities.

Chris     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.