Let's go over this geometry problem step by step.

The figure shows a rectangle with a right triangle removed from it, and you need to find the area of the shaded region.

Given:

• The dimensions of the rectangle are 10 ft by 7 ft.
• The dimensions of the right triangle are a base of 6 ft and a height of 3 ft.

Steps to solve:

1. Find the area of the rectangle: $\text{Area of rectangle} = \text{length} \times \text{width} = 10 \, \text{ft} \times 7 \, \text{ft} = 70 \, \text{ft}^2$

2. Find the area of the triangle: $\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \, \text{ft} \times 3 \, \text{ft} = 9 \, \text{ft}^2$

3. Find the shaded area: $\text{Shaded area} = \text{Area of rectangle} - \text{Area of triangle} = 70 \, \text{ft}^2 - 9 \, \text{ft}^2 = 61 \, \text{ft}^2$

So, the area of the shaded region is 61 ft².

Let me know if you need further clarification on any part of the solution!

Additional questions for understanding:

1. What would the area be if the triangle was a different shape (e.g., an isosceles triangle)?
2. How would the area change if the base of the triangle were increased to 8 ft?
3. If the rectangle’s length was reduced by 2 feet, how would that affect the shaded area?
4. Can you find the perimeter of the shaded region?
5. What is the significance of subtracting the area of the triangle from the rectangle?

Tip: When dealing with areas of composite shapes (like rectangles with triangles removed), always break the shape into simpler parts (like the rectangle and triangle here), find the area of each part, and then combine them as needed.