To solve this problem, we need to find the area of the shaded region in the triangle after removing the rectangle.

Step 1: Calculate the area of the right triangle.

The area $A$ of a right triangle is given by:

$A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height}$

In this case, the base of the triangle is 10 yards and the height is 9 yards. So,

$A_{\text{triangle}} = \frac{1}{2} \times 10 \times 9 = 45 \text{ square yards}$

Step 2: Calculate the area of the rectangle.

The area $A$ of the rectangle is given by:

$A_{\text{rectangle}} = \text{length} \times \text{width}$

From the figure, the length of the rectangle is 6 yards and the width is 3 yards. So,

$A_{\text{rectangle}} = 6 \times 3 = 18 \text{ square yards}$

Step 3: Calculate the area of the shaded region.

The shaded region is the area of the triangle minus the area of the rectangle:

$A_{\text{shaded}} = A_{\text{triangle}} - A_{\text{rectangle}} = 45 - 18 = 27 \text{ square yards}$

Thus, the area of the shaded region is 27 square yards.

Would you like any more details or have any further questions?

Here are some related questions you could explore:

1. How would the area change if the dimensions of the rectangle were different?
2. How can you generalize this approach for any triangle and rectangle dimensions?
3. What if the rectangle was placed in a different position within the triangle—would the result change?
4. How would the area be affected if the triangle were an isosceles triangle instead of a right triangle?
5. What happens to the area if we change the orientation of the triangle?

Tip: Always ensure the dimensions provided are correctly identified for base and height when dealing with triangles, especially in more complex shapes.