Problem Analysis
The problem involves a triangular plot of land, specifically triangle \( \triangle ABC \), and the farmer plans to sell a smaller triangular plot \( \triangle DEC \). We need to determine how much land the farmer will still own after selling \( \triangle DEC \).
#### Given:
1. \( \triangle ABC \) is a right triangle with:
- \( AB = 2 \) km (height).
- \( AC = 4 \) km (base, since \( D \) is the midpoint of \( AC \) and \( AD = DC = 2 \) km).
2. \( \triangle DEC \) is a smaller right triangle with:
- \( DE = 1 \) km (height).
- \( DC = 2 \) km (base).
#### Objective:
Calculate the area of \( \triangle ABC \) and subtract the area of \( \triangle DEC \) to find the remaining land area.
---
Step-by-Step Solution
#### 1. Calculate the area of \( \triangle ABC \)
The formula for the area of a right triangle is:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
For \( \triangle ABC \):
- Base \( AC = 4 \) km.
- Height \( AB = 2 \) km.
\[
\text{Area of } \triangle ABC = \frac{1}{2} \times 4 \times 2 = 4 \text{ km}^2
\]
#### 2. Calculate the area of \( \triangle DEC \)
For \( \triangle DEC \):
- Base \( DC = 2 \) km.
- Height \( DE = 1 \) km.
\[
\text{Area of } \triangle DEC = \frac{1}{2} \times 2 \times 1 = 1 \text{ km}^2
\]
#### 3. Determine the remaining land area
The remaining land area is the area of \( \triangle ABC \) minus the area of \( \triangle DEC \):
\[
\text{Remaining area} = \text{Area of } \triangle ABC - \text{Area of } \triangle DEC
\]
\[
\text{Remaining area} = 4 \text{ km}^2 - 1 \text{ km}^2 = 3 \text{ km}^2
\]
---
Final Answer
The farmer will still own \( 3 \) square kilometers of land after selling the smaller plot \( \triangle DEC \).
\[
\boxed{3}
\]
Parent Tip: Review the logic above to help your child master the concept of free sat math worksheet.
