Let’s solve each part of this problem.

Given Information:

• $ABCD$ is a rectangle inscribed in a circle.
• The area of triangle $\triangle BCD$ is $24 \, \text{cm}^2$.
• There is a circular road of $2 \, \text{cm}$ around the circle.

Problem Breakdown:

We’ll solve this in two steps:

1. Find the radius of the circle (small circle around rectangle ABCD).
2. Find the area of the road around the small circle.

Step-by-Step Solution:

Part (a): Find the Radius of the Circle

1. Triangle Area Relation: Since the area of $\triangle BCD$ is given as $24 \, \text{cm}^2$, we can use this information to find the sides of rectangle $ABCD$.

2. Diagonals of the Rectangle: The diagonal $BD$ of rectangle $ABCD$ will also be the diameter of the circle in which it is inscribed. We need to calculate the length of $BD$.

3. Using Area of Triangle $\triangle BCD$: Suppose $BC$ is one side and $CD$ is the other side of rectangle $ABCD$. Let:

   • $BC = x$ (one side of the rectangle)
   • $CD = y$ (other side of the rectangle)

   Since the area of $\triangle BCD = \frac{1}{2} \times x \times y$, $\frac{1}{2} \times x \times y = 24$ $x \times y = 48$

4. Diameter of the Circle: The diameter $BD$ of the circle can be calculated using the Pythagorean theorem: $BD = \sqrt{x^2 + y^2}$

   To find the values of $x$ and $y$ individually, we need additional information or assumptions about the rectangle, which might be missing here. If the rectangle were a square, we would solve for $x = y$. However, without knowing more, we’ll assume we can find $x$ and $y$.

Part (b): Find the Area of the Road Around the Small Circle

If we have the radius $r$ of the small circle, then the radius of the large circle (including the road) will be $r + 2 \, \text{cm}$.

1. Area of the Small Circle (using radius $r$): $A_{\text{small}} = \pi r^2$

2. Area of the Large Circle (using radius $r + 2$): $A_{\text{large}} = \pi (r + 2)^2$

3. Area of the Road: The area of the road is the difference between the area of the large circle and the area of the small circle: $A_{\text{road}} = A_{\text{large}} - A_{\text{small}}$ $A_{\text{road}} = \pi (r + 2)^2 - \pi r^2$ $A_{\text{road}} = \pi \left((r + 2)^2 - r^2\right)$

4. Expanding the Equation: $A_{\text{road}} = \pi \left(r^2 + 4r + 4 - r^2\right)$ $A_{\text{road}} = \pi (4r + 4)$ $A_{\text{road}} = 4\pi (r + 1)$

To complete the solution, we would need the exact value of $r$ (radius of the small circle) based on the dimensions of the rectangle.

Would you like a detailed example with assumed values, or do you have any further questions about this setup?

Related Questions

1. How would the solution change if we knew the rectangle $ABCD$ was a square?
2. Can we solve this problem if the area of another triangle within $ABCD$ was given?
3. What is the significance of inscribing a rectangle in a circle in geometry?
4. How would we calculate the radius if the area of the entire rectangle was given instead?
5. Could we solve this using trigonometry if the angles within the rectangle were given?

Tip:

Always start by writing down all known values and relationships. This helps clarify any assumptions or missing information needed for the solution.