Find the area of the quadrilateralthe coordinates of whose angular points taken in order are (1,1), (3,4),(5,-2) and (4,-7)

Options

A. 10 sq. units
B. 41 sq. units
C. 82 sq. units
D. 61.5 sq. units

Answer

To find the area of the quadrilateral with vertices at the coordinates (1,1), (3,4), (5,-2), and (4,-7), we can divide the quadrilateral into two triangles and calculate the area of each triangle using the formula for the area of a triangle given its vertices.

### Step-by-Step Solution:

1. **Identify the vertices of the quadrilateral:**
- Let the points be A(1,1), B(3,4), C(5,-2), and D(4,-7).

2. **Divide the quadrilateral into two triangles:**
- We can divide the quadrilateral into triangles ABC and ACD.

3. **Calculate the area of triangle ABC:**
- The formula for the area of a triangle given vertices (x1, y1), (x2, y2), (x3, y3) is:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
- For triangle ABC:
- A(1,1), B(3,4), C(5,-2)
- Substitute the coordinates into the formula:
\[ \text{Area}_{ABC} = \frac{1}{2} \left| 1(4 - (-2)) + 3((-2) - 1) + 5(1 - 4) \right| \]
\[ = \frac{1}{2} \left| 1(6) + 3(-3) + 5(-3) \right| \]
\[ = \frac{1}{2} \left| 6 - 9 - 15 \right| \]
\[ = \frac{1}{2} \left| -18 \right| = \frac{18}{2} = 9 \]

4. **Calculate the area of triangle ACD:**
- For triangle ACD:
- A(1,1), C(5,-2), D(4,-7)
- Substitute the coordinates into the formula:
\[ \text{Area}_{ACD} = \frac{1}{2} \left| 1(-2 - (-7)) + 5((-7) - 1) + 4(1 - (-2)) \right| \]
\[ = \frac{1}{2} \left| 1(5) + 5(-8) + 4(3) \right| \]
\[ = \frac{1}{2} \left| 5 - 40 + 12 \right| \]
\[ = \frac{1}{2} \left| -23 \right| = \frac{23}{2} = 11.5 \]

5. **Calculate the total area of the quadrilateral:**
- Total area = Area of triangle ABC + Area of triangle ACD
\[ \text{Total Area} = 9 + 11.5 = 20.5 \]

### Final Answer:
The area of the quadrilateral is **20.5 square units**.