Question

two altitudes of a scalene triangle have length 4 and 12.if the length of remaining also an integer ,then it's maximum value is

Answer 1

The inequality that connects the altitudes of triangle is derived from the inequality that connects the sides of a triangle.

If a1 & a2a1 & a2 are two altitudes of a triangle, then the third altitude a3a3 of the triangle must be bounded by the inequality:

11a1+1a2<a3<11a1−1a211a1+1a2<a3<11a1−1a2

We have : a1=4 & a2=12a1=4 & a2=12

So, 114+112<a3<114−112114+112<a3<114−112

113<a3<116113<a3<116

3<a3<63<a3<6

Therefore, the longest altitude possible in the triangle is 5 cm (Considering the integral value)