Triangle Inequality
Lesson · Beginner
Geometry: Plane Geometry
The three inequalities are equivalent to
and similarly,
Two sides of a triangle have lengths 7 and 11. Find the possible range of the third side x.
Therefore,
If x is an integer,
A triangle has integer side lengths, one of which is 20. Its perimeter is 50. How many different unordered triples of side lengths are possible?
Let the other sides be a and b with
Then
We need the three numbers a, b, 20 to form a triangle.
Since
that inequality is automatic.
We also need
Using
gives
Also,
so
Therefore,
giving
different unordered triangles.
For any nondegenerate polygon, each side is shorter than the sum of all the other sides.
For an n-gon with side lengths
we have
A pentagon has integer side lengths 3, 5, 7, 9, x. Find all positive integer values of x for which a nondegenerate pentagon can exist.
The largest side must be less than the sum of the other four.
If
then x is the largest side, so we need
Thus
If
then 9 is the largest side.
We need
which is automatically true for every positive x.
Therefore,
Let . These numbers are the altitudes of some triangle if and only if their reciprocals satisfy the triangle inequalities:
For any triangle of area ,
Therefore satisfy the triangle inequalities exactly when
do.
And conversely, if the three reciprocals satisfy the triangle inequalities, construct a triangle whose sides are proportional to
Then scale that triangle appropriately; its altitudes can be made exactly .