 |
| ||
| |||
|
| 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:
- 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
- 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.