Find the area of the quadrilateral whose vertices are A(-4,5), B(0,7), C(5,-5) and D(-4,-2).

Explanation

To find the area of the quadrilateral with vertices A(-4,5), B(0,7), C(5,-5), and D(-4,-2), we can divide the quadrilateral into two triangles and calculate their areas separately.

### Step-by-Step Solution:

1. **Identify the vertices of the quadrilateral:**
- A(-4, 5)
- B(0, 7)
- C(5, -5)
- D(-4, -2)

2. **Divide the quadrilateral into two triangles:**
- Triangle 1: A, B, D
- Triangle 2: B, C, D

3. **Calculate the area of Triangle ABD using the formula:**
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Here, \( (x_1, y_1) = A(-4, 5) \), \( (x_2, y_2) = B(0, 7) \), \( (x_3, y_3) = D(-4, -2) \).

Substituting the coordinates:
\[ \text{Area}_{ABD} = \frac{1}{2} \left| -4(7 - (-2)) + 0((-2) - 5) + (-4)(5 - 7) \right| \]
\[ = \frac{1}{2} \left| -4(9) + 0 + (-4)(-2) \right| \]
\[ = \frac{1}{2} \left| -36 + 8 \right| \]
\[ = \frac{1}{2} \left| -28 \right| = \frac{1}{2} \times 28 = 14 \]

4. **Calculate the area of Triangle BCD using the same formula:**
Here, \( (x_1, y_1) = B(0, 7) \), \( (x_2, y_2) = C(5, -5) \), \( (x_3, y_3) = D(-4, -2) \).

Substituting the coordinates:
\[ \text{Area}_{BCD} = \frac{1}{2} \left| 0(-5 - (-2)) + 5((-2) - 7) + (-4)(7 - (-5)) \right| \]
\[ = \frac{1}{2} \left| 0 + 5(-9) + (-4)(12) \right| \]
\[ = \frac{1}{2} \left| 0 - 45 - 48 \right| \]
\[ = \frac{1}{2} \left| -93 \right| = \frac{93}{2} = 46.5 \]

5. **Calculate the total area of the quadrilateral ABCD:**
\[ \text{Area}_{ABCD} = \text{Area}_{ABD} + \text{Area}_{BCD} = 14 + 46.5 = 60.5 \]

### Final Answer:
The area of the quadrilateral ABCD is **60.5 square units**.