Question

#16 listen 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. perimeter: □ ft area: □ ft²

Explanation:

Step1: Analyze the shape (L - shaped, can be split or use perimeter trick)

The figure is a polygon with vertices \( D(1,1) \), \( C(8,1) \), \( B(8,7) \), \( A(5,7) \), \( F(5,4) \), \( E(1,4) \), back to \( D(1,1) \). For perimeter, we can use the "rectangle perimeter" trick (since horizontal and vertical sides: the horizontal length from \( x = 1 \) to \( x = 8 \) is \( 8 - 1 = 7 \) units, vertical from \( y = 1 \) to \( y = 7 \) is \( 7 - 1 = 6 \) units. Wait, no, actually, the L - shape's perimeter is equal to the perimeter of a rectangle with length \( 8 - 1 = 7 \) units and height \( 7 - 1 = 6 \) units? Wait, let's check side lengths:

- \( DC \): from \( (1,1) \) to \( (8,1) \): length \( 8 - 1 = 7 \) units.
- \( CB \): from \( (8,1) \) to \( (8,7) \): length \( 7 - 1 = 6 \) units.
- \( BA \): from \( (8,7) \) to \( (5,7) \): length \( 8 - 5 = 3 \) units.
- \( AF \): from \( (5,7) \) to \( (5,4) \): length \( 7 - 4 = 3 \) units.
- \( FE \): from \( (5,4) \) to \( (1,4) \): length \( 5 - 1 = 4 \) units? Wait no, \( 5 - 1 = 4 \)? Wait \( x \) from 1 to 5: \( 5 - 1 = 4 \)? Wait \( E(1,4) \) to \( F(5,4) \): \( 5 - 1 = 4 \) units.
- \( ED \): from \( (1,4) \) to \( (1,1) \): length \( 4 - 1 = 3 \) units? Wait no, \( y \) from 4 to 1: \( 4 - 1 = 3 \)? Wait no, \( 4 - 1 = 3 \)? Wait \( (1,4) \) to \( (1,1) \): \( 4 - 1 = 3 \) units? Wait, no, let's recalculate correctly:

Wait, maybe a better way: the L - shape can be thought of as two rectangles or as a rectangle with a smaller rectangle removed, but for perimeter, the horizontal and vertical segments:

Horizontal sides:
- Bottom: \( DC \): 7 units (from \( x = 1 \) to \( x = 8 \), \( y = 1 \))
- Top: \( BA \) (3 units, \( x = 5 \) to \( x = 8 \), \( y = 7 \)) and \( FE \) (4 units, \( x = 1 \) to \( x = 5 \), \( y = 4 \)) → total horizontal top: \( 3 + 4 = 7 \) units (same as bottom)
Vertical sides:
- Right: \( CB \): 6 units (from \( y = 1 \) to \( y = 7 \), \( x = 8 \))
- Left: \( ED \): 3 units (from \( y = 1 \) to \( y = 4 \), \( x = 1 \)) and \( AF \): 3 units (from \( y = 4 \) to \( y = 7 \), \( x = 5 \)) → total vertical left: \( 3 + 3 = 6 \) units (same as right)

So perimeter in units: \( 2\times(7 + 6)=26 \) units. Since each unit is 100 feet, perimeter is \( 26\times100 = 2600 \) feet.

Step2: Calculate Area

Split the L - shape into two rectangles:
- Rectangle 1: \( DCFE \): from \( (1,1) \) to \( (8,1) \) to \( (8,4) \) to \( (1,4) \)? Wait no, better:
- Rectangle 1: \( D(1,1) \), \( C(8,1) \), \( F(5,4) \), \( E(1,4) \)? No, split into lower rectangle ( \( D(1,1) \), \( C(8,1) \), \( (8,4) \), \( (1,4) \)) and upper rectangle ( \( A(5,7) \), \( B(8,7) \), \( C(8,4) \), \( F(5,4) \))? Wait, coordinates:
Lower rectangle: \( D(1,1) \), \( C(8,1) \), \( (8,4) \), \( E(1,4) \): length \( 8 - 1 = 7 \) units, height \( 4 - 1 = 3 \) units. Area: \( 7\times3 = 21 \) square units.
Upper rectangle: \( A(5,7) \), \( B(8,7) \), \( C(8,4) \), \( F(5,4) \): length \( 8 - 5 = 3 \) units, height \( 7 - 4 = 3 \) units. Area: \( 3\times3 = 9 \) square units.
Total area: \( 21 + 9 = 30 \) square units. Since each unit is 100 feet, 1 square unit is \( 100\times100 = 10000 \) square feet. So area is \( 30\times10000 = 300000 \) square feet.

Wait, let's check another way: using the "big rectangle minus small rectangle". The big rectangle would be from \( (1,1) \) to \( (8,7) \): length \( 7 \) units, height \( 6 \) units. Area: \( 7\times6 = 42 \) square units. The small rectangle missing is from \( (1,4) \) to \( (5,7) \): length \( 5 - 1 = 4 \) units, height \( 7 - 4 = 3 \) units. Area: \( 4\times3 = 12 \) square units. So total area: \( 42 - 12 = 30 \) square units. Same as before. Good.

Answer:

Perimeter: \(\boldsymbol{2600}\) ft
Area: \(\boldsymbol{300000}\) \( \text{ft}^2 \)