Question

Find the area of the quadrilateral whose vertices are \( (1,1) \), \( (3,4) \), \( (5,-2) \) and \( (4,-7) \) taken in order.

Hint:

For polygons whose vertices are given in order, the shoelace formula is the fastest method to find the area directly from coordinates.

Answer 1

Step 1: Write the coordinates in tabular form for using the shoelace formula.}
The given vertices of the quadrilateral taken in order are \[ (1,1), \quad (3,4), \quad (5,-2), \quad (4,-7) \] Now, we write them in order and repeat the first point at the end: \[ \begin{array}{c|c} x & y
\hline 1 & 1
3 & 4
5 & -2
4 & -7
1 & 1 \end{array} \]
Step 2: Find the sum of the products of downward diagonals.}
Using the shoelace rule, we calculate \[ S_1 = (1 \times 4) + (3 \times -2) + (5 \times -7) + (4 \times 1) \] \[ S_1 = 4 - 6 - 35 + 4 = -33 \]
Step 3: Find the sum of the products of upward diagonals.}
Now, \[ S_2 = (1 \times 3) + (4 \times 5) + (-2 \times 4) + (-7 \times 1) \] \[ S_2 = 3 + 20 - 8 - 7 = 8 \]
Step 4: Apply the formula for area of a quadrilateral.}
The area of the quadrilateral is given by \[ \text{Area} = \frac{1}{2}\left|S_1 - S_2\right| \] Substituting the values, we get \[ \text{Area} = \frac{1}{2}\left|-33 - 8\right| \] \[ \text{Area} = \frac{1}{2}\left|-41\right| = \frac{41}{2} \]
Step 5: Write the final answer.}
Therefore, the area of the quadrilateral is \[ \boxed{\frac{41}{2} \text{ square units}} \] or \[ \boxed{20.5 \text{ square units}} \]