One theorem that tells us about the relationship between the sides of a triangle is the Triangle Inequality Theorem. To understand this theorem, start with a picture of a triangle, such as triangle ABC.
Triangle Inequality Theorem: Any side of a triangle must be shorter than the sum of the lengths of the other two sides.
If you apply this theorem to the triangle above, you can say that the inequalities below are true.
AC < AB + BC
BC < AB + AC
AB < BC + AC
At first glance, the Triangle Inequality Theorem may seem a bit strange. You may recall from a previous lesson that by definition, a theorem is a conjecture (a statement based on the observation of a pattern) that has been proven true. Let's test this theorem out by first looking at some examples. Study the example on each of the tabs below, looking for evidence that theorem holds true.
Example 1
Example 2
Example 3
Example 4
See how the length of side C of the triangle must ALWAYS be less than the lengths of sides A and B put together? The same is true for the other sides. Side B will be less than A + C. Side A will be less than B + C.
This triangle instrument also follows the Triangle Inequality Theorem in that any side of the triangle (A, B , or C) will ALWAYS be less than the sum of the other two sides. For the triangle instrument, A < B + C; B < A + C; C < A + B
You may think that the bottom side looks pretty long, right? Still, the bottom side (C ) will always be smaller than the top two sides of the hanger (A and B) added together. This holds true for sides A and B as well.
Here's is an attempt to build a triangle with the bottom side (5) NOT LESS than the other two sides added together (3 + 2). If you try to draw a triangle that defies the Triangle Inequality Theorem, this is the kind of shape you get. See how the side length 3 and side length 2 are too short to form a real triangle? If they were longer, their sum would be greater than 5, and they would be able to meet and form a triangle.
