Find the value of "a", and the area of the shaded region.

Let's solve the problem step by step.

### Given:
- Radius of the circle \( r = 5 \) meters.
- A rectangle is inscribed in the circle, with one side \( 7 \) meters long (height of the rectangle), and the other side as \( a \) meters (the base of the rectangle).

### Step 1: Relationship Between \( a \) and \( r \)
Since the rectangle is inscribed in the circle, the diagonal of the rectangle equals the diameter of the circle. 

The diameter \( D \) of the circle is:
\[
D = 2r = 2 \times 5 = 10 \text{ meters}
\]

Using the Pythagorean theorem for the rectangle, where the diagonal is the hypotenuse:
\[
\text{Diagonal}^2 = a^2 + 7^2
\]
But since the diagonal is equal to the diameter:
\[
10^2 = a^2 + 7^2
\]
\[
100 = a^2 + 49
\]
\[
a^2 = 100 - 49 = 51
\]
\[
a = \sqrt{51} \approx 7.14 \text{ meters}
\]

### Step 2: Area of the Shaded Region
The area of the shaded region is the area of the circle minus the area of the rectangle.

#### Area of the Circle:
\[
\text{Area of the circle} = \pi r^2 = \pi \times 5^2 = 25\pi \text{ square meters}
\]

#### Area of the Rectangle:
\[
\text{Area of the rectangle} = 7 \times a = 7 \times \sqrt{51} \approx 7 \times 7.14 = 49.98 \text{ square meters}
\]

#### Area of the Shaded Region:
\[
\text{Shaded area} = 25\pi - 49.98 \approx 78.54 - 49.98 = 28.56 \text{ square meters}
\]

### Final Answers:
- The value of \( a \) is approximately \( 7.14 \) meters.
- The area of the shaded region is approximately \( 28.56 \) square meters.

Let me know if you need further clarification or details!

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Here are some additional related questions you might find interesting:

1. What would be the value of \( a \) if the radius of the circle was 6 meters?
2. How would the area of the shaded region change if the rectangle had a height of 8 meters instead of 7?
3. What is the perimeter of the rectangle in this configuration?
4. Can you derive a general expression for the area of the shaded region in terms of the circle's radius and one side of the rectangle?
5. What if the circle's radius was halved? How would that affect the area of the shaded region?

**Tip:** Always ensure that when dealing with inscribed figures, the Pythagorean theorem is a powerful tool for relating different dimensions.

Explore how to find the value of 'a' and the area of the shaded region in this circle geometry problem. Step-by-step solution includes the application of the Pythagorean theorem and circle geometry principles. Learn about inscribed rectangles, circle diameter relationships, and geometric calculations.

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