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

To solve the problem, we need to determine the value of \( a \) and the area of the shaded region in the circle.

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

### Step 1: Relationship Between \( a \) and \( r \)
The diagonal of the rectangle is the diameter of the circle since the rectangle is inscribed in the circle. The diameter \( D \) of the circle is \( 2r = 10 \) meters.

Using the Pythagorean theorem on the rectangle, we have:
\[
\text{Diagonal}^2 = a^2 + 7^2
\]
But the diagonal is also the diameter, so:
\[
10^2 = a^2 + 7^2
\]
\[
100 = a^2 + 49
\]
\[
a^2 = 100 - 49 = 51
\]
\[
a = \sqrt{51} \approx 7.14 \text{ meters}
\]

### 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.

Would you like any further details or explanations?

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Here are some related questions for further practice:

1. What would be the area of the shaded region if the radius was 6 meters?
2. How would the value of \( a \) change if the height of the rectangle was 8 meters?
3. Can you derive a general formula for the area of the shaded region in terms of \( r \) and \( a \)?
4. What is the perimeter of the rectangle in this problem?
5. If the area of the shaded region was given, how could you find the value of \( r \)?

**Tip:** Understanding how the Pythagorean theorem applies to inscribed shapes is essential when solving geometry problems involving circles and rectangles.

Learn how to solve a geometry problem involving finding the value of 'a' and calculating the area of the shaded region in a circle. This problem uses the Pythagorean theorem and involves calculating areas of geometric shapes. Suitable for high school students.

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