**Problem Analysis:**

In this problem, we are given a figure where a goat is tied to a pole at point O, which is the center of a semicircular plot. We need to find the length of the rope through which the goat is tied to the pole.

Let's analyze the given information step by step.

**Step 1: Understanding the figure:**

We have a semicircular plot where the center of the plot is point O. The goat is tied to the pole at point O.

**Step 2: Understanding the problem statement:**

The problem states that the area of the plot is equal to its perimeter. We need to find the length of the rope through which the goat is tied to the pole.

**Step 3: Understanding the formulae:**

To solve this problem, we need to understand the formulas for the area and perimeter of a circle.

- The area of a circle is given by the formula: A = πr^2, where A is the area and r is the radius of the circle.
- The perimeter of a circle is given by the formula: P = 2πr, where P is the perimeter and r is the radius of the circle.

**Step 4: Solving the problem:**

Let's assume that the radius of the semicircular plot is r and the length of the rope through which the goat is tied to the pole is BO.

To find the area of the plot, we need to find the area of the semicircle. The area of the semicircle is half of the area of the circle. So, the area of the semicircle is A = (πr^2)/2.

To find the perimeter of the plot, we need to find the perimeter of the semicircle and the length of the rope. The perimeter of the semicircle is half of the perimeter of the circle plus the length of the rope. So, the perimeter of the semicircle is P = (2πr)/2 + BO.

According to the problem statement, the area of the plot is equal to its perimeter. So, we have the equation (πr^2)/2 = (2πr)/2 + BO.

Simplifying this equation, we get BO = (πr^2)/2 - πr.

Hence, the length of the rope through which the goat is tied to the pole is (πr^2)/2 - πr.

**Conclusion:**

In conclusion, the length of the rope through which the goat is tied to the pole is (πr^2)/2 - πr, where r is the radius of the semicircular plot.