Determine a method to calculate the third side of a triangle using only the lengths of two known sides | Step-by-Step Solution

1. What We're Solving:

We want to find the length of the third side of a triangle when we only know the lengths of two sides, without using any angle measurements or trigonometric functions.

2. The Approach:

We can't determine a unique answer with only two sides! If you have two sticks of fixed lengths, you can arrange them at many different angles to form different triangles, each with a different third side length.

We CAN determine the range of possible values for that third side using the Triangle Inequality Theorem. This is actually more useful than you might think - it tells us what's geometrically possible!

3. Step-by-Step Solution:

Step 1: Understand the Triangle Inequality The Triangle Inequality states that the sum of any two sides of a triangle must be greater than the third side. This gives us three conditions for sides a, b, and c:

• a + b > c
• a + c > b
• b + c > a

Step 2: Set up your known information Let's say you know sides a and b, and you want to find the possible values for side c.

Step 3: Apply the constraints From the Triangle Inequality, we get:

• From a + b > c: This means c < a + b
• From a + c > b: This means c > b - a (rearranging: c + a > b becomes c > b - a)
• From b + c > a: This means c > a - b

Step 4: Combine the constraints The third side c must satisfy: |a - b| < c < a + b

The absolute value |a - b| represents the larger of (a - b) or (b - a), ensuring we get a positive lower bound.

4. The Answer:

For a triangle with known sides of length a and b, the third side c must satisfy: |a - b| < c < a + b

This means c is greater than the absolute difference of the two known sides, and less than their sum.

Example: If you have sides of length 5 and 8:

• Lower bound: |5 - 8| = 3, so c > 3
• Upper bound: 5 + 8 = 13, so c < 13
• Therefore: 3 < c < 13

5. Memory Tip:

Remember it as the "difference and sum rule": The unknown side must be more than the difference and less than the sum of the two known sides. It's like the triangle is "stretching" between these two extremes - it can't be so small that the sides don't meet, or so large that they can't bend to connect!

This is a beautiful example of how geometry gives us boundaries and possibilities rather than just single answers. Great question! 🌟