1:CA - Chapter \( 4> \) Section Exercises \( 44 \geq \) Evercise 24 In a topcoraphical map of a cib, the vertioes of the city linits are \( A(10,9), B(18,9), C(18,2), D(14,4.5), E(10,4.5) \). The ccordinates are measured in miles What is the area of the city? The area of the cily is 12 square miles.

To find the area of the city defined by the vertices \( A(10,9) \), \( B(18,9) \), \( C(18,2) \), \( D(14,4.5) \), and \( E(10,4.5) \), we can use the shoelace formula (or surveyor's formula) for calculating the area of a polygon given its vertices. ### Step 1: List the coordinates We will list the coordinates in order, closing the polygon by repeating the first vertex at the end: \[ \begin{align*} A & : (10, 9) \\ B & : (18, 9) \\ C & : (18, 2) \\ D & : (14, 4.5) \\ E & : (10, 4.5) \\ A & : (10, 9) \quad \text{(repeat the first vertex)} \end{align*} \] ### Step 2: Apply the shoelace formula The shoelace formula for the area \( A \) of a polygon with vertices \((x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)\) is given by: \[ A = \frac{1}{2} \left| \sum_{i=1}^{n} (x_i y_{i+1} - y_i x_{i+1}) \right| \] Where \( (x_{n+1}, y_{n+1}) \) is the same as \( (x_1, y_1) \). ### Step 3: Calculate the sums Let's calculate the sums for our vertices: \[ \begin{align*} \text{Sum 1} & : x_1y_2 + x_2y_3 + x_3y_4 + x_4y_5 + x_5y_1 \\ & = 10 \cdot 9 + 18 \cdot 2 + 18 \cdot 4.5 + 14 \cdot 4.5 + 10 \cdot 9 \\ & = 90 + 36 + 81 + 63 + 90 = 360 \end{align*} \] \[ \begin{align*} \text{Sum 2} & : y_1x_2 + y_2x_3 + y_3x_4 + y_4x_5 + y_5x_1 \\ & = 9 \cdot 18 + 9 \cdot 18 + 2 \cdot 14 + 4.5 \cdot 10 + 4.5 \cdot 10 \\ & = 162 + 162 + 28 + 45 + 45 = 442 \end{align*} \] ### Step 4: Calculate the area Now we can substitute these sums into the shoelace formula: \[ A = \frac{1}{2} \left| 360 - 442 \right| = \frac{1}{2} \left| -82 \right| = \frac{1}{2} \cdot 82 = 41 \] ### Conclusion The area of the city is \( 41 \) square miles, not \( 12 \) square miles as stated. If you have any further questions or need additional assistance, feel free to ask!