Problem 2

Explanation

Let $F$ be the center of the circle, so $AF = FD = FG = r$ (the radius of the circle).

Given the side length of the square $AD = 24 \text{ cm}$. Since $E$ is the midpoint of $AD$ (based on the symmetry of the circle passing through $A$ and $D$), then:

Finding the Radius of the Circle

We will find the radius of the circle ($r$) using the Pythagorean theorem on the right-angled triangle $\triangle AEF$. Note that the length $EF$ is the distance from the center of the circle to side $AD$. Since the circle is tangent to side $BC$ at point $G$, and the total distance from side $AD$ to side $BC$ is the side length of the square ($24 \text{ cm}$), then $EF = AB - r = 24 - r$.

Using the Pythagorean theorem on $\triangle AEF$:

Calculating the Area of the Circle

After finding the radius $r = 15 \text{ cm}$, we can calculate the area:

Therefore, the area of the circle is $225\pi \text{ cm}^2$.