Question

Find the area of the polygon.

Figure QRSTU is shown. Q is at 3, 0. R is at negative 3, 0. S is at negative 3, 7. T is at 1, 11. U is at 5, 7.

59 square units
65 square units
72 square units
83 square units

Answer 1

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:
1. \( Q(3, 0) \)
2. \( R(-3, 0) \)
3. \( S(-3, 7) \)
4. \( T(1, 11) \)
5. \( U(5, 7) \)
6. 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:

1. 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: \( 0 - 21 - 33 + 7 + 0 = -47 \).

2. 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.