To find the area of the polygon QRSTU given the vertices \( Q(3, 0) \), \( R(-3, 0) \), \( S(-3, 7) \), \( T(1, 11) \), and \( U(5, 7) \), we can use the shoelace formula (or surveyor's formula).
The shoelace formula states that for a polygon with vertices at \( (x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n) \), the area \( A \) 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 considered to be \( (x_1, y_1) \).
Let's list down the vertices in order:
- \( Q(3, 0) \)
- \( R(-3, 0) \)
- \( S(-3, 7) \)
- \( T(1, 11) \)
- \( U(5, 7) \)
- Returning to \( Q(3, 0) \)
Now, set up the coordinates for use in the formula:
\[ \begin{array}{c|c} x & y \ \hline 3 & 0 \ -3 & 0 \ -3 & 7 \ 1 & 11 \ 5 & 7 \ 3 & 0 \ \end{array} \]
Now apply the shoelace formula:
\[ A = \frac{1}{2} \left| (3 \cdot 0 + (-3) \cdot 7 + (-3) \cdot 11 + 1 \cdot 7 + 5 \cdot 0) - (0 \cdot (-3) + 0 \cdot (-3) + 7 \cdot 1 + 11 \cdot 5 + 7 \cdot 3) \right| \]
Calculating each term:
-
For \( x_i y_{i+1} \):
- \( 3 \cdot 0 = 0 \)
- \( (-3) \cdot 7 = -21 \)
- \( (-3) \cdot 11 = -33 \)
- \( 1 \cdot 7 = 7 \)
- \( 5 \cdot 0 = 0 \)
Summing these: \( 0 - 21 - 33 + 7 + 0 = -47 \).
-
For \( y_i x_{i+1} \):
- \( 0 \cdot (-3) = 0 \)
- \( 0 \cdot (-3) = 0 \)
- \( 7 \cdot 1 = 7 \)
- \( 11 \cdot 5 = 55 \)
- \( 7 \cdot 3 = 21 \)
Summing these: \( 0 + 0 + 7 + 55 + 21 = 83 \).
Putting this together:
\[ A = \frac{1}{2} \left| -47 - 83 \right| = \frac{1}{2} \left| -130 \right| = \frac{130}{2} = 65 \]
Thus, the area of the polygon QRSTU is \( \boxed{65} \) square units.
1 answer