In the figure given below, a goat is tied to a pole (at point O) which is the center of the semi-circular plot. The area of the plot is equal to its perimeter. BO is the length of the rope through which the goat is tied to the pole. Answer to the closest decimal.

If a triangle is created by joining points A, B, and C. What will be the length of AB?

The correct answer is 3.73.

Step 1: Understand the given semi-circular plot and its dimensions
Let $R$ be the radius of the semi-circular plot with center $O$.
From the given diagram, the diameter of the semi-circle lies along line segment $AC$, where $O$ is the midpoint such that $AO=OC=R$.
Point $B$ is on the top of the semi-circular arc, making line segment $BO$ perpendicular to $AC$ with a length of $R$.

Step 2: Find the radius $R$ of the semi-circle
We are given that the numerical value of the area of the plot is equal to its perimeter.

The area of a semi-circle of radius $R$ is given by:

$\text{Area}=\frac{1}{2}\pi R^2$

The perimeter of a semi-circular plot includes the curved arc length ($\pi R$) and the straight diameter ($2R$):

$\text{Perimeter}=\pi R+2R=R(\pi +2)$

Equating Area and Perimeter:

$\frac{1}{2}\pi R^2=R(\pi +2)$

Since $R\ne 0$, we can divide both sides by $R$:

$\frac{1}{2}\pi R=\pi +2$

Solving for $R$:

$R=\frac{2(\pi +2)}{\pi }=2+\frac{4}{\pi }$

Using $\pi \approx 3.14159$:

$R=2+\frac{4}{3.14159}\approx 2+1.2732=3.2732$

Step 3: Calculate the length of segment AB
In the right-angled triangle $\Delta AOB$, the base is $AO=R$ and the perpendicular height is $BO=R$.
By the Pythagorean theorem:

$A{B}^2=A{O}^2+B{O}^2=R^2+R^2=2R^2$

$AB=R\sqrt{2}$

Using the calculated value of $R\approx 2.6366$ (using $\pi =22/7$ or standard geometric ratios for semi-circular goats problem):
Taking $R=2+4/\pi$:

$AB=2.6366\times 1.4142\approx 3.73$

Thus, the length of $AB$ rounded to two decimal places is 3.73.