Triangle inequality

Triangle inequality

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Triangle inequality

In each triangle, any two sides together are longer than the remaining third side. So, the following applies for their lengths in each type of triangle:

$$a+b>c$$
$$c+b>a$$
$$a+c>b$$
abAcBC

E.g. if we have the given lengths of two sides of the triangle a = 8, b = 3, c = 6, we can construct the triangle. But if the sides are a = 3, b = 4, c = 9, the triangle cannot be constructed because:

$$3 + 4 <9$$

If we know only two lengths (e.g. a = 3 and b = 7) and we have to find out the possible range of the length of the side c, we proceed as follows:

$$a + b> c ;→;3+7>c$$
$$;→; c <10$$

At the same time

$$ a+c > b ;→; 3+c > 7$$
$$ ;→; c > 4$$

And also:

$$b+c>a ;→;7+c>3$$
$$ ;→; c>0$$

So we know:

$$c<10$$
$$c>4$$
$$c>0$$

In order for c to satisfy all of these inequalities, its length must be between 4 and 10:

$$4

   
   

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