A triangle cannot be constructed from any three random line segments. Attempting to do so may result in the creation of a degenerate triangle, a triangle that fails to have all of its sides connect. To ensure that we get a legitimate triangle, we use something known as the Triangle Inequality, which relates the three sides of the triangle. It states:
If \(a\), \(b\), and \(c\) are the lengths of the three sides of a triangle, then
$$a + b > c$$ $$a + c > b$$ $$b + c > a$$
If any one of these inequalities is not true, then we get a degenerate triangle. We will show basic calculations involving these formulas, including determining if a triangle is degenerate.
Here will be given all three side lengths of a triangle, and you must determine whether the triangle is degenerate.
Example 1: Is a triangle with sides of lengths \(5\), \(8\), and \(7\) a degenerate triangle?
Solution: Let \(a = 5\), \(b = 8\), and \(c = 7\). To verify that this is not a degenerate triangle, we must plug these side lengths into all three parts of the Triangle Inequality:
$$5 + 8 > 7$$ $$5 + 7 > 8$$ $$7 + 8 > 5$$
All three of these inequalities are true, so this is not a degenerate triangle.
Example 2: Is a triangle with sides of lengths \(3\), \(4\), and \(8\) degenerate?
Solution: One side is longer than the other two combined:
$$3 + 4 < 8$$
Therefore a degenerate triangle is produced. Even though
$$4 + 8 > 3$$ $$3 + 8 > 4$$
all three inequalities must be true, and only two of them are.
Example 3: Do the side lengths \(3\), \(4\), and \(7\) produce a degenerate triangle?
Solution: Here, notice that one side as exactly as long as the other two combined, so \(3 + 4 = 7\), and it is not true that \(3 + 4 > 7\), which violates the Triangle Inequality, so this triangle is degenerate.
In this section of the article we will use examples to illustrate how to construct triangles so that they are not degenerate.
Example 4: Two sides of a triangle have lengths of \(2\) and \(4\). What is the range of possible lengths for the third side?
Solution: Let \(c\) be the side with an unknown length. Then all three inequalities must be satisfied by the Triangle Inequality:
$$2 + c > 4$$ $$4 + c > 2$$ $$2 + 4 > c$$
Isolate \(c\) in each inequality and simplify as much as possible:
$$c > 2$$ $$c > -2$$ $$6 > c$$
From these restrictions we know that \(c\) is greater than \(2\) but less than \(6\).
Example 5: Equilateral triangles are triangles where all three sides have the same length. Prove that these triangles are not degenerate.
Solution: Let the side length of all three triangles be \(a\). Then, by the Triangle Inequality, we plug the side lengths into the three conditions of the formula:
$$a + a > a$$ $$a + a > a$$ $$a + a > a$$
Since \(a\) must be positive (since side lengths are always positive), this inequality is true. As a result, an equilateral triangle will never be degenerate.
Example 6: An isosceles triangle has two sides of the same length. Prove that the third side is less than twice as long as either of the sides of the same length.
Solution: Let the side length of one of the isosceles sides be \(s\). Let the side length of the third side be \(t\). Then by the Triangle Inequality:
$$s + s > t$$
Simplify the left side:
$$2s > t$$
This inequality states that the combined length of the two sides of the same length is longer than the length of the other side, exactly what we wanted to prove.
The Triangle Inequality is a simple concept that relates the sides of a triangle. It serves as a solid check that you have not made a degenerate triangle, and it is a fundamental law at the base of geometry.