FENCING Capria is planning to fence a triangular plot of land. Two of the sides of the plot measure 230 yards and 490 yards.

1. Break Down the Problem

To find the total length of fencing needed, we need to determine the third side of the triangular plot. Since only two sides are given, we cannot find the exact length of the third side without more information.

2. Relevant Concepts

In a triangle, the lengths of the sides must satisfy the triangle inequality theorem, which states:

• The sum of the lengths of any two sides must be greater than the length of the third side.

Let's denote the sides as follows:

• Side $a = 230$ yards
• Side $b = 490$ yards
• Side $c = ?$ (the length of the third side)

3. Analysis and Detail

Using the triangle inequality theorem, we can derive the following inequalities based on the two known sides $a$ and $b$:

1. $a + b > c$ becomes $230 + 490 > c$

   • $720 > c$ or $c < 720$

2. $a + c > b$ becomes $230 + c > 490$

   • $c > 490 - 230$
   • $c > 260$

3. $b + c > a$ becomes $490 + c > 230$

   • This inequality is always true since $c$ is always positive.

From the above inequalities, we have:

• $260 < c < 720$

4. Verify and Summarize

The third side $c$ can have any length greater than 260 yards and less than 720 yards to maintain the properties of a triangle. Without a specific length given for the third side, we cannot calculate the total fencing length precisely. However, the fencing required would be at least $230 + 490 + 260 = 980$ yards (if $c$ is at its minimum) and less than $230 + 490 + 720 = 1440$ yards (if $c$ is at its maximum).

Final Answer

The total length of fencing required is between 980 yards and 1440 yards, depending on the length of the third side $c$.