We are given a rectangle $ABCD$ inscribed in a semicircle with diameter along $CD$. The dimensions of the rectangle are $AB = 6$ and $BC = 7$, where $AB$ is the height of the rectangle and $BC$ is one of the sides parallel to the diameter.

Step 1: Understanding the Geometry

• The semicircle's diameter coincides with $CD$, so the diameter of the semicircle is equal to the length of $CD$.
• The center of the semicircle is the midpoint of $CD$, and the radius of the semicircle is half the diameter.

Since the rectangle is inscribed in the semicircle, the diagonal of the rectangle $AC$ must be the radius of the semicircle.

Step 2: Applying the Pythagorean Theorem

We can calculate the length of the diagonal $AC$ using the Pythagorean theorem. The diagonal is the hypotenuse of a right triangle with legs $AB = 6$ and $BC = 7$.

$AC = \sqrt{AB^2 + BC^2} = \sqrt{6^2 + 7^2} = \sqrt{36 + 49} = \sqrt{85}$

Thus, the length of the diagonal $AC$ is $\sqrt{85}$. Since the diagonal equals the radius of the semicircle, the radius of the semicircle is $\sqrt{85}$.

Step 3: Calculating the Area of the Semicircle

The formula for the area of a semicircle is:

$\text{Area of semicircle} = \frac{1}{2} \pi r^2$

Substitute $r = \sqrt{85}$ into the formula:

$\text{Area of semicircle} = \frac{1}{2} \pi (\sqrt{85})^2 = \frac{1}{2} \pi \cdot 85 = \frac{85}{2} \pi$

Step 4: Solving for $a$

We are told that the area of the semicircle is $a\pi$. From the above expression, we can see that:

$a = \frac{85}{2}$

Thus, the value of $a$ is $\boxed{\frac{85}{2}}$.

Would you like more details on any specific part of this solution? Here are 5 related questions to explore further:

1. How would the solution change if the rectangle had different dimensions?
2. What is the relationship between the radius and the diagonal of the rectangle?
3. Can you derive the formula for the area of a semicircle?
4. How would this problem change if the figure were a full circle instead of a semicircle?
5. What if the rectangle was rotated inside the semicircle?

Tip: The Pythagorean theorem is a powerful tool for finding relationships between the sides of right triangles, and it often comes up when working with geometric figures inscribed in circles.