MODELING REAL LIFE  The diagram shows the vertices of a lion sanctuary. Each unit in the coordinate plane represents 100 feet. Find the perimeter and the area of the sanctuary.

Coordinate plane shows vertices of lion sanctuary A B C D E. The vertical and horizontal axis extends from 0 to 8. The coordinates of polygon A B C D E. Ordered pair A 5 comma 7. Ordered pair B 8 comma 7. Ordered pair C 8 comma 1. Ordered pair D 1 comma 1. Ordered pair E 1 comma 4. Ordered pair F 5 comma 4.
Perimeter:  ft

Area:  ft2

To find the perimeter and area of the lion sanctuary defined by the vertices \( A(5, 7) \), \( B(8, 7) \), \( C(8, 1) \), \( D(1, 1) \), \( E(1, 4) \), and \( F(5, 4) \), we can follow these steps:

### Step 1: Calculate the Perimeter
To find the perimeter, we need to calculate the length of each side of the polygon and sum them up.

1. **Side AB**: 
   - Coordinates: \( A(5, 7) \) and \( B(8, 7) \)
   - Distance: \( \sqrt{(8 - 5)^2 + (7 - 7)^2} = \sqrt{3^2 + 0^2} = 3 \) units
2. **Side BC**:
   - Coordinates: \( B(8, 7) \) and \( C(8, 1) \)
   - Distance: \( \sqrt{(8 - 8)^2 + (7 - 1)^2} = \sqrt{0^2 + 6^2} = 6 \) units
3. **Side CD**:
   - Coordinates: \( C(8, 1) \) and \( D(1, 1) \)
   - Distance: \( \sqrt{(1 - 8)^2 + (1 - 1)^2} = \sqrt{7^2 + 0^2} = 7 \) units
4. **Side DE**:
   - Coordinates: \( D(1, 1) \) and \( E(1, 4) \)
   - Distance: \( \sqrt{(1 - 1)^2 + (4 - 1)^2} = \sqrt{0^2 + 3^2} = 3 \) units
5. **Side EF**:
   - Coordinates: \( E(1, 4) \) and \( F(5, 4) \)
   - Distance: \( \sqrt{(5 - 1)^2 + (4 - 4)^2} = \sqrt{4^2 + 0^2} = 4 \) units
6. **Side FA**:
   - Coordinates: \( F(5, 4) \) and \( A(5, 7) \)
   - Distance: \( \sqrt{(5 - 5)^2 + (7 - 4)^2} = \sqrt{0^2 + 3^2} = 3 \) units

The perimeter is the sum of all these distances:

\[
\text{Perimeter} = 3 + 6 + 7 + 3 + 4 + 3 = 26 \text{ units}
\]

Each unit represents 100 feet, so the perimeter in feet is:

\[
\text{Perimeter} = 26 \times 100 = 2600 \text{ feet}
\]

### Step 2: Calculate the Area
To calculate the area, we'll use the **Shoelace Theorem** (also known as the **Surveyor's formula**).

\[
\text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_4 + x_4y_5 + x_5y_6 + x_6y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_5 + y_5x_6 + y_6x_1) \right|
\]

Substitute the coordinates of the vertices \( A(5,7) \), \( B(8,7) \), \( C(8,1) \), \( D(1,1) \), \( E(1,4) \), \( F(5,4) \):

\[
\text{Area} = \frac{1}{2} \left| 5(7) + 8(1) + 8(1) + 1(4) + 1(4) + 5(7) - [7(8) + 7(8) + 1(1) + 1(1) + 4(5) + 4(5)] \right|
\]

Simplify the calculations:

\[
\text{Area} = \frac{1}{2} \left| 35 + 8 + 8 + 4 + 4 + 35 - (56 + 56 + 1 + 1 + 20 + 20) \right|
\]

\[
\text{Area} = \frac{1}{2} \left| 94 - 154 \right| = \frac{1}{2} \times 60 = 30 \text{ square units}
\]

Since each unit represents 100 feet, each square unit represents \(100 \times 100 = 10,000\) square feet. So, the area in square feet is:

\[
\text{Area} = 30 \times 10,000 = 300,000 \text{ square feet}
\]

### Final Answers:
- **Perimeter**: \( 2600 \) feet
- **Area**: \( 300,000 \) square feet

Would you like more details, or do you have any questions?

Here are five related questions:
1. How does the Shoelace Theorem work for different polygon shapes?
2. What is the difference between perimeter and area?
3. How would the calculations change if the units were in meters instead of feet?
4. Can the Shoelace Theorem be used for a polygon with more than six sides?
5. How can the perimeter and area formulas be used in real-life applications, like construction?

**Tip:** Always double-check coordinates and calculations to avoid errors in geometry problems.

Learn how to calculate the perimeter and area of a lion sanctuary using coordinate geometry. Detailed step-by-step solution with distances calculated using the distance formula and area computed using the Shoelace Theorem. Suitable for high school students.

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