Determining the Range of Possible Measures for the Third Side of a Triangle

When working with triangles, one of the most fundamental principles is the concept of the triangle inequality theorem. This theorem helps us determine the possible lengths of the third side given the lengths of the other two sides. In this article, we will explore a specific problem where the lengths of two sides of a triangle are given as 4 and 10 units, and we will determine the range of possible lengths for the third side.

Introduction to the Triangle Inequality Theorem

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Mathematically, if the three sides of a triangle are labeled a, b, and c, then the following conditions must be satisfied:

a b c b c a c a b

Let's apply this theorem to the problem at hand.

Solving the Problem Using the Triangle Inequality Theorem

Given two sides of a triangle with lengths a 4 units and b 10 units, we need to find the range of possible lengths for the third side c.

Using the triangle inequality theorem, we can set up the following inequalities:

4 10 c ? c 14 4 c 10 ? c 6 10 c 4 (This condition is always satisfied since c must be positive)

Combining the first two inequalities, we get:

6 c 14

This means the range of possible measures for the third side is 6 c 14.

Additional Calculations Using Cosine Law

Fascinatingly, the cosine law can also be used to further explore the possible lengths of the third side. The cosine law states:

c^2 a^2 b^2 - 2ab cosθ

where θ is the angle between sides a and b.

When θ 0°

If the angle between the sides is 0°, the cosine of 0° is 1, and the triangle would essentially become a straight line, which is not a valid triangle. However, for the sake of illustration:

c^2 5^2 11^2 - 2(5)(11)cos0° c^2 25 121 - 110 c^2 36 c 6

When θ 180°, the triangle becomes a degenerate triangle (a straight line), but for the sake of our illustration:

c^2 5^2 11^2 - 2(5)(11)cos180° c^2 25 121 110 c^2 256 c 16

When θ 90°, the triangle becomes a right triangle:

c^2 5^2 11^2 - 2(5)(11)cos90° c^2 25 121 - 0 c^2 146 c √146 ≈ 12.08

Therefore, the range of possible lengths for the third side, using the cosine law, is 6 ≤ c ≤ 12.08.

Conclusion

In conclusion, the range of possible measures for the third side of a triangle with sides 4 units and 10 units is between 6 and 14 units. Using the cosine law, we can refine this range to between 6 and 12.08 units. This demonstrates the practical application of the triangle inequality theorem and the cosine law in solving geometric problems.