How Short Can It Be?

Suppose two sides of a triangle are 6 and 10, and the third side x is not known yet. If 10 is the longest side, the triangle inequality says that x and 6 together must add to more than 10:

x + 6 > 10

x > 4

Any value of x above 4 passes this test, for example 4.1, 5 or 9. At exactly 4 the triangle is flat. Below 4 the two shorter sides are too short to meet.

Once you have the inequality, solve it the way you solved inequalities before: subtract the known side, 6, from both sides. The inequality comes from one rule of geometry, that two sides together must add to more than the third side. After that, it is ordinary algebra. This gives the lower limit on x. The next lesson gives the upper limit.