The triangle inequality theorem is fundamental in geometry. It states that for any triangle with sides a, b, and c, the sum of any two sides must be greater than the third side. This means we need three conditions: a plus b greater than c, a plus c greater than b, and b plus c greater than a. Let's see an example with sides of length 2, 3, and 4.5 units. We check: 2 plus 3 equals 5, which is greater than 4.5, so these sides can form a triangle.
Now let's construct a valid triangle using compass and straightedge. We have three sides with lengths 5, 7, and 9 units. First, we check the triangle inequality: 5 plus 7 equals 12, which is greater than 9, so these sides can form a triangle. We start by drawing the base side of length 9. Then we use a compass set to radius 7 to draw an arc from point A, and another compass set to radius 5 to draw an arc from point B. The intersection of these arcs gives us point C, completing our triangle construction.
Now let's see what happens when we try to construct a triangle that violates the triangle inequality. We have sides of length 3, 4, and 10 units. Let's check: 3 plus 4 equals 7, which is less than 10. This violates the triangle inequality. When we attempt the construction by drawing the base of length 10 and using compass arcs of radius 3 and 4, we see that the arcs don't intersect. There's a gap between them, making it impossible to complete the triangle. This demonstrates why the triangle inequality must be satisfied.
Let's explore how changing one side length affects triangle formation dynamically. We fix two sides at lengths 3 and 4, and gradually increase the third side c. Initially, when c is small, the compass arcs intersect easily and we can form a triangle. As c increases, the arcs still intersect but the triangle becomes more stretched. However, when c reaches 7, which equals 3 plus 4, the arcs just touch at one point. Beyond this critical value, the arcs no longer intersect, making triangle formation impossible.
For a triangle to exist, all three triangle inequality conditions must be satisfied simultaneously. Let's examine a triangle with sides 6, 8, and 10. First condition: 6 plus 8 equals 14, which is greater than 10 - check. Second condition: 6 plus 10 equals 16, which is greater than 8 - check. Third condition: 8 plus 10 equals 18, which is greater than 6 - check. Since all three conditions are satisfied, we can successfully construct this triangle using compass and straightedge.