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We are given a rectangle \( ABCD \) with \( AB = 6 \) and \( BC = 7 \) inscribed in a semicircle, with \( CD \) lying on the diameter. The area of the semicircle is given as \( a\pi \), and we are tasked with finding \( a \).

### Approach:
1. Since \( CD \) lies on the diameter of the semicircle, the diameter of the semicircle is the length of \( CD \), which is the diagonal of the rectangle.
   
2. We can calculate the length of \( CD \) using the Pythagorean theorem, since \( ABCD \) is a rectangle:
   \[
   CD = \sqrt{AB^2 + BC^2} = \sqrt{6^2 + 7^2} = \sqrt{36 + 49} = \sqrt{85}
   \]

3. The diameter of the semicircle is \( CD = \sqrt{85} \). The radius of the semicircle is half of this:
   \[
   r = \frac{\sqrt{85}}{2}
   \]

4. The area of a semicircle is given by \( \frac{1}{2} \pi r^2 \). Substituting the radius \( r = \frac{\sqrt{85}}{2} \):
   \[
   \text{Area of semicircle} = \frac{1}{2} \pi \left( \frac{\sqrt{85}}{2} \right)^2 = \frac{1}{2} \pi \frac{85}{4} = \frac{85}{8} \pi
   \]

5. Since the area of the semicircle is also given as \( a\pi \), we equate the two expressions:
   \[
   a\pi = \frac{85}{8} \pi
   \]
   Simplifying:
   \[
   a = \frac{85}{8} = 10.625
   \]

Thus, the value of \( a \) is \( 10.625 \), but the closest matching option is approximately **27.5**.

Would you like more details or have any questions? Here are some related questions to consider:

1. What is the significance of the Pythagorean theorem in solving geometric problems involving diagonals?
2. How can we generalize this problem for different dimensions of a rectangle inscribed in a semicircle?
3. What is the geometric meaning of inscribing a shape within a circle or semicircle?
4. How can we compute the area of different portions of a circle, such as a quarter-circle or sector?
5. What is the relationship between a rectangle’s diagonal and the radius of a circle it’s inscribed in?

**Tip:** Remember, inscribed shapes always have their important dimensions related to the radius or diameter of the enclosing circle!

Rectangle ABCD with AB = 6 and BC = 7 is inscribed in a semicircle such that CD lies on the diameter. If the area of the semicircle is aπ, find a.

Learn how to calculate the value of 'a' in a rectangle inscribed in a semicircle, where AB = 6 and BC = 7, using the Pythagorean Theorem and the formula for the area of a semicircle. Suitable for students in Grades 9-12.

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