To solve this problem, we need to calculate the area of the triangular land and then subtract the area of the trapezoidal parcel that Mr. Smith sold to his brother. Here’s how we can approach this:

Step 1: Calculate the Area of the Triangle

The triangular plot has a base of $200 \, \text{ft}$ and a height of $250 \, \text{ft}$. The area $A$ of a triangle can be found using the formula: $A = \frac{1}{2} \times \text{base} \times \text{height}$ Substituting in the given values: $A = \frac{1}{2} \times 200 \times 250 = 100 \times 250 = 25000 \, \text{square feet}$ So, the area of the triangular land is $25000 \, \text{square feet}$.

Step 2: Calculate the Area of the Trapezoidal Parcel

The trapezoidal parcel is shown with two parallel sides (the top and bottom lengths). The top side is $80 \, \text{ft}$, the bottom side is $100 \, \text{ft}$, and the height of the trapezoid (the distance between these parallel sides) is $50 \, \text{ft}$.

The area $A$ of a trapezoid is calculated using the formula: $A = \frac{1}{2} \times (\text{base}_1 + \text{base}_2) \times \text{height}$ Substituting in the values for the trapezoidal parcel: $A = \frac{1}{2} \times (80 + 100) \times 50 = \frac{1}{2} \times 180 \times 50 = 90 \times 50 = 4500 \, \text{square feet}$ So, the area of the trapezoidal parcel is $4500 \, \text{square feet}$.

Step 3: Calculate the Retained Land Area

To find the area of the land Mr. Smith retains, subtract the area of the trapezoidal parcel from the area of the triangular land: $\text{Retained Area} = 25000 - 4500 = 20500 \, \text{square feet}$

Final Answer

The area of the land Mr. Smith retains is: $\boxed{20500 \, \text{square feet}}$

Would you like more details on any part of this solution, or do you have any questions?

Here are some additional questions related to this problem:

1. What would be the area if Mr. Smith decided to sell only half of the trapezoidal parcel?
2. How would the retained area change if the height of the triangle was increased to 300 feet?
3. If the triangle was divided into two equal parts, what would be the area of each part?
4. How could we calculate the perimeter of the triangular piece of land?
5. What other shapes could be created by dividing the triangular land differently?

Tip: When working with composite areas, break down the shape into simpler parts (like triangles or trapezoids) to make calculations easier.