Determining the Length of the Third Side in a Triangle Given Two Sides

A common question in geometry involves finding the length of the third side of a triangle when the lengths of two sides are known. However, the problem is not as simple as it might appear at first glance. Let's explore the complexities and how to determine the possible lengths of the third side accurately.

Triangle Inequality Theorem

The Triangle Inequality Theorem states that in any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. This theorem is crucial in determining the possible lengths of the third side.

Given two sides of a triangle with lengths 3 and 4, let's denote these sides as a 3 and b 4. The third side, denoted as c, must satisfy the following conditions:

c 3 > 4 c 4 > 3 3 4 > c

Simplifying these inequalities, we get:

c > 1 c > -1 (which is always true since c is a positive length) c

Therefore, the third side c must be greater than 1 and less than 7. This means that the possible lengths for the third side are any values in the range (1, 7).

Ambiguous Triangle

It is important to note that the problem statement does not provide the measure of the included angle between the two given sides. Without this information, it is impossible to find a unique value for the third side. However, the Law of Cosines can provide a range of possible values for c based on the angle A between sides a and b.

Law of Cosines

The Law of Cosines is a powerful tool for finding the length of the third side of a triangle when the lengths of the other two sides and the included angle are known. The formula is given by:

c2 a2 b2 - 2ab cosA

Using the given values a 3 and b 4, we can rewrite the equation as:

c2 32 42 - 2(3)(4) cosA c2 9 16 - 24 cosA c2 25 - 24 cosA

Depending on the value of cosA, the value of c2 will vary, and thus the value of c will also vary. However, since the maximum value of cosA is 1 and the minimum value is -1, the range of c2 is:

When cosA 1, c2 25 - 24 1, so c 1 When cosA -1, c2 25 24 49, so c 7

Therefore, the possible values for c based on the Law of Cosines are in the range (1, 7).

Summary

In conclusion, the possible lengths of the third side of a triangle given the two sides of lengths 3 and 4 are any values in the range (1, 7). This is consistent with the Triangle Inequality Theorem and the Law of Cosines. Without the measure of the included angle, no specific value for the third side can be determined.