The coordinate plane shows the floor plan for a swimming pool. What is the area of the pool’s border?
A graph plots the distance of the floor plan of a swimming pool with a pool and pool border. The data points of the pool are (3, 12), (7, 15), (13, 7) and (9, 4) and pool border are (0, 12), (9, 18), (17, 6) and (9, 0).
A.
50 square meters
B.
65 square meters
C.
80 square meters
D.
100 square meters
E.
125 square meters
To find the area of the pool's border, we need to calculate the area of the polygon formed by connecting the points (3, 12), (7, 15), (13, 7), (9, 4), (0, 12), (9, 18), (17, 6), and (9, 0) in order.
First, let's plot the points on the coordinate plane:
```
(0,12) (9,18) (17,6)
+-----------+
| |
| |
| |
+-----------+
(3,12) (9,4)
```
```
(9,0)
|
|
|
|
(7,15) (13,7)
```
Now, let's label the points as A, B, C, D, E, F, G, and H clockwise starting from (3, 12):
```
D(0,12) E(9,18) F(17,6)
A+-----------+B
| |
| |
| |
+-----------+
C(3,12) H(9,4)
```
```
H(9,0)
|
|
|
|
G(7,15) F(13,7)
```
Next, let's calculate the length of each side of the polygon:
AB: distance between (3, 12) and (7, 15)
AB = sqrt((7-3)^2 + (15-12)^2) = sqrt(4^2 + 3^2) = 5
BC: distance between (7, 15) and (13, 7)
BC = sqrt((13-7)^2 + (7-15)^2) = sqrt(6^2 + (-8)^2) = 10
CD: distance between (13, 7) and (9, 4)
CD = sqrt((9-13)^2 + (4-7)^2) = sqrt((-4)^2 + (-3)^2) = 5
DE: distance between (9, 4) and (0, 12)
DE = sqrt((0-9)^2 + (12-4)^2) = sqrt((-9)^2 + 8^2) = 10
EF: distance between (0, 12) and (9, 18)
EF = sqrt((9-0)^2 + (18-12)^2) = sqrt(9^2 + 6^2) = 3√5
FG: distance between (9, 18) and (17, 6)
FG = sqrt((17-9)^2 + (6-18)^2) = sqrt(8^2 + (-12)^2) = 4√10
GH: distance between (17, 6) and (9, 0)
GH = sqrt((9-17)^2 + (0-6)^2) = sqrt((-8)^2 + (-6)^2) = 10
HA: distance between (9, 0) and (7, 15)
HA = sqrt((7-9)^2 + (15-0)^2) = sqrt((-2)^2 + 15^2) = √229
Now, we can use the Shoelace Formula to calculate the area of the polygon:
Area = 1/2 * | (3 * 10) + (10 * 3√5) + (5 * 4√10) + (4√10 * √229) + (√229 * 10) + (10 * 5) + (5 * 3) + (3 * 12)
Area = 1/2 * | 30 + 30√5 + 20√10 + 4√229 + 10√229 + 50 + 15 + 36 |
Area = 1/2 * | 100√229 + 121 + 105√5 |
Area = 1/2 * (100√229 + 121 + 105√5)
Using a calculator, the approximate value of the area is 670.96 square units.
Therefore, the closest option to the area of the pool's border is 670.96 square units is:
E. 125 square meters.