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To calculate the total length of the fence Emma needs (i.e., the perimeter of the shape shown in the grid), we need to calculate the lengths of each side of the polygon formed by the points \( A \), \( B \), \( C \), and \( D \).

The coordinates of the vertices are:
- \( A(0, 5) \)
- \( B(4, 5) \)
- \( C(4, 2) \)
- \( D(-2, 2) \)

### Step 1: Calculate the distance between points using the distance formula.
The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

#### Calculate \( AB \) (distance between points \( A(0, 5) \) and \( B(4, 5) \)):
\[
AB = \sqrt{(4 - 0)^2 + (5 - 5)^2} = \sqrt{16 + 0} = 4 \, \text{units}
\]

#### Calculate \( BC \) (distance between points \( B(4, 5) \) and \( C(4, 2) \)):
\[
BC = \sqrt{(4 - 4)^2 + (2 - 5)^2} = \sqrt{0 + 9} = 3 \, \text{units}
\]

#### Calculate \( CD \) (distance between points \( C(4, 2) \) and \( D(-2, 2) \)):
\[
CD = \sqrt{(-2 - 4)^2 + (2 - 2)^2} = \sqrt{36 + 0} = 6 \, \text{units}
\]

#### Calculate \( DA \) (distance between points \( D(-2, 2) \) and \( A(0, 5) \)):
\[
DA = \sqrt{(0 - (-2))^2 + (5 - 2)^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.61 \, \text{units}
\]

### Step 2: Calculate the total perimeter.
\[
\text{Perimeter} = AB + BC + CD + DA = 4 + 3 + 6 + 3.61 = 16.61 \, \text{units}
\]

### Conclusion:
The closest value to the calculated perimeter of 16.61 units is not listed directly, but based on available options, the nearest value would be **18.74 units**.

Do you have any questions or would you like further details? Here are some additional related questions:

1. How would the perimeter change if one of the vertices was shifted?
2. What formula is used to calculate the distance between two points in 3D space?
3. How do you find the area of a polygon if the perimeter is already known?
4. Can you calculate the perimeter of a circular fence instead of a polygonal one?
5. How does changing the grid's scale affect the perimeter calculation?

**Tip:** Always double-check your coordinate points before calculating distances on a grid!

Emma is trying to build a fence around her backyard as shown on the coordinate grid. She needs to calculate the total length of the fence required. Which value is closest to the perimeter of the fencing Emma needs?

Learn how to calculate the total length of a fence required to enclose a backyard using the distance formula in coordinate geometry. This geometry problem is suitable for students in Grades 9-10 and covers essential concepts such as the distance formula and calculating the perimeter of a polygon on a grid.

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