If the area of the triangle with vertices at the points whose coordinates are (2, 5), (0, 3) and (4, k) is 4 units, then find the value of k.

Answer

To find the value of \( k \) such that the area of the triangle with vertices at the points \( (2, 5) \), \( (0, 3) \), and \( (4, k) \) is 4 units, we can use the formula for the area of a triangle given by its vertices.

### Step-by-Step Solution:

1. **Identify the vertices**:
Let the vertices of the triangle be:
- \( A(2, 5) \)
- \( B(0, 3) \)
- \( C(4, k) \)

2. **Area formula**:
The area \( A \) of a triangle with vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is given by:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Substituting the coordinates of points \( A \), \( B \), and \( C \):
\[ A = \frac{1}{2} \left| 2(3 - k) + 0(k - 5) + 4(5 - 3) \right| \]

3. **Simplify the expression**:
\[ A = \frac{1}{2} \left| 2(3 - k) + 0 + 4(2) \right| \]
\[ A = \frac{1}{2} \left| 2(3 - k) + 8 \right| \]
\[ A = \frac{1}{2} \left| 6 - 2k + 8 \right| \]
\[ A = \frac{1}{2} \left| 14 - 2k \right| \]

4. **Set the area equal to 4**:
Since the area is given as 4 units:
\[ \frac{1}{2} \left| 14 - 2k \right| = 4 \]

5. **Multiply both sides by 2**:
\[ \left| 14 - 2k \right| = 8 \]

6. **Solve the absolute value equation**:
This gives us two cases:
- Case 1: \( 14 - 2k = 8 \)
- Case 2: \( 14 - 2k = -8 \)

**For Case 1**:
\[ 14 - 2k = 8 \]
\[ -2k = 8 - 14 \]
\[ -2k = -6 \]
\[ k = 3 \]

**For Case 2**:
\[ 14 - 2k = -8 \]
\[ -2k = -8 - 14 \]
\[ -2k = -22 \]
\[ k = 11 \]

7. **Final values of \( k \)**:
The possible values of \( k \) are \( 3 \) and \( 11 \).

### Conclusion:
The values of \( k \) that satisfy the area condition are \( k = 3 \) or \( k = 11 \).