Fundamental Ideas
Triangles
In Δ
TAB (Figure
1 ), if
T, A, and
B represent three points on a map and you want to go from
T to
B, going from
T to
A to
B would obviously be longer than going directly from
T to
B. The following theorem expresses this idea.
|
|
|||
|
|||
|
|
|||
|
|||
|
|
Theorem 38 (Triangle Inequality Theorem): The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Example 1: In Figure
2 , the measures of two sides of a triangle are 7 and 12. Find the range of possibilities for the third side.
|
|
|||
|
|||
|
|
|||
|
|||
|
|
Using the Triangle Inequality Theorem, you can write the following:
7 + x > 12, so x > 5
7 + 12 > x, so 19 > x (or x < 19)
Therefore, the third side must be more than 5 and less than 19.
