Finding The Third Side Of A Triangle Given 2

Finding the Third Side of a Triangle: A Comprehensive Guide

Determining the length of the third side of a triangle when you know the lengths of two sides requires more information than just the two sides themselves. A triangle is defined by three sides and three angles, and knowing only two sides leaves the third side with a range of possible values. This article will explore the different scenarios and techniques used to find the third side, focusing on the crucial role of the angle between the known sides.

Understanding the Triangle Inequality Theorem

Before delving into the methods, it's crucial to understand a fundamental concept in geometry: the Triangle Inequality Theorem. This 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:

• a + b > c
• a + c > b
• b + c > a

where 'a', 'b', and 'c' represent the lengths of the three sides. This theorem provides constraints on the possible values of the third side. If the given side lengths violate this theorem, then a triangle with those dimensions cannot exist.

Scenario 1: Knowing Two Sides and the Included Angle (SAS)

This is the most common and straightforward scenario. If you know two sides of a triangle and the angle between them (Side-Angle-Side, or SAS), you can use the Law of Cosines to find the third side.

The Law of Cosines

The Law of Cosines is a powerful tool that relates the lengths of the sides of a triangle to the cosine of one of its angles. The formula is:

c² = a² + b² - 2ab * cos(C)

Where:

• 'a' and 'b' are the lengths of the two known sides.
• 'C' is the angle between sides 'a' and 'b'.
• 'c' is the length of the third side (the one you want to find).

Example:

Let's say you have a triangle with sides a = 5, b = 7, and the included angle C = 60°. To find the length of side 'c', we plug the values into the Law of Cosines:

c² = 5² + 7² - 2 * 5 * 7 * cos(60°) c² = 25 + 49 - 70 * 0.5 c² = 74 - 35 c² = 39 c = √39 ≈ 6.24

Therefore, the length of the third side is approximately 6.24.

Scenario 2: Knowing Two Sides and a Non-Included Angle (SSA)

This scenario, also known as the ambiguous case, is more complex because it can lead to multiple solutions, one solution, or no solution at all. The reason for this ambiguity lies in the fact that we don't know the relative positions of the sides.

Solving the Ambiguous Case (SSA)

To solve for the third side in the SSA case, we use the Law of Sines. However, before applying it, we need to consider the possible solutions:

• No Solution: If the sine of the angle opposite the unknown side is greater than 1, there's no triangle that can be formed with the given dimensions. This violates the fundamental properties of sine function.

• One Solution: This occurs when the given side opposite the known angle is shorter than the other known side and the sine of the opposite angle equals 1. Or if the conditions do not meet the criteria for a two-solution case.

• Two Solutions: This occurs when the given side opposite the known angle is shorter than the other known side and the height of the triangle calculated using the given information is less than the known side opposite the given angle, and there are two possible triangles that can satisfy the given conditions.

The Law of Sines:

The Law of Sines states:

a/sin(A) = b/sin(B) = c/sin(C)

Where:

• 'a', 'b', and 'c' are the lengths of the sides.
• 'A', 'B', and 'C' are the angles opposite those sides.

Example (Two Solutions):

Let's assume we have a = 8, b = 10, and A = 30°. Using the Law of Sines:

8/sin(30°) = 10/sin(B) sin(B) = 10 * sin(30°) / 8 sin(B) ≈ 0.625 B ≈ 38.68° or B ≈ 141.32°

This shows that two angles could satisfy the given dimensions. Each of these values for 'B' would lead to a different triangle and a different length for side 'c' using the Law of Sines to solve for c. The possibility of two solutions arises from the fact that two different triangles can share the same two sides and one of the angles.

Example (One Solution):

Suppose we modify the above example: a = 12, b = 10, and A = 30°. Solving for B using the Law of Sines would only give one possible solution as the other solution for B would be greater than 180° - A - B and hence, create an impossible triangle.

Example (No Solution):

Consider a scenario where a = 5, b = 10, and A = 80°. Upon applying the Law of Sines, the resulting sine value would be greater than 1, which is not possible. Therefore, no triangle would exist.

Scenario 3: Knowing Two Sides and the Angle Opposite One of Them (SSA - Alternative Approach)

In the ambiguous case (SSA), you can also use a graphical approach. Draw a line segment representing one of the known sides (let's say side 'b'). At one end, draw an angle equal to the known angle (angle B). Then, draw an arc with a radius equal to the length of the other known side (side 'a'). If this arc intersects the line drawn from the angle, you have a solution (or possibly two). If not, there is no solution. This method helps visualize the possibilities and can be helpful to grasp the ambiguous case intuitively.

Scenario 4: Right-Angled Triangles

If the triangle is a right-angled triangle (one angle is 90°), finding the third side is simpler using the Pythagorean theorem:

a² + b² = c²

where 'c' is the hypotenuse (the side opposite the right angle).

Practical Applications

Understanding how to find the third side of a triangle has numerous applications in various fields:

• Surveying: Determining distances and land areas.
• Navigation: Calculating distances and directions.
• Engineering: Designing structures and calculating forces.
• Physics: Solving problems in mechanics and kinematics.
• Computer Graphics: Creating realistic three-dimensional models.

Conclusion

Finding the third side of a triangle hinges on the information available. The most straightforward case involves knowing two sides and the included angle (SAS), where the Law of Cosines provides a direct solution. The ambiguous case (SSA) requires careful consideration of the Law of Sines and the potential for multiple solutions or no solution at all. Understanding the Triangle Inequality Theorem is essential in all scenarios to ensure the existence of a valid triangle. Mastering these techniques is fundamental for anyone working in fields that require geometric calculations. Remember to always check your answers to ensure they are physically feasible and realistic within the context of the problem. Employing both algebraic solutions and graphical visualization can enhance understanding and assist in accurate problem-solving.