Question

1. graph quadrilateral efgh
2. find the length of ef
3. find the length of fg
4. find the length of gh
5. find the length of eh
6. add all 4 side lengths to find the perimeter of efgh

Explanation:

To solve the problem of finding the lengths of the sides and the perimeter of quadrilateral \( EFGH \), we need to assume the coordinates of the vertices (since the graph is provided, we'll infer typical grid coordinates). Let's assume the coordinates (for a standard grid) as follows (we'll define them to proceed):

Step 1: Define Coordinates (Assumed from Grid)

Let’s assume the coordinates of the vertices (estimating from the grid):

• \( E(x_1, y_1) \)
• \( F(x_2, y_2) \)
• \( G(x_3, y_3) \)
• \( H(x_4, y_4) \)

For example, if we assume:

• \( E(-2, -1) \)
• \( F(-2, 5) \)
• \( G(4, 3) \)
• \( H(4, -1) \)

Step 2: Length of \( EF \)

\( EF \) is a vertical line (same \( x \)-coordinate). The length is the difference in \( y \)-coordinates.
\[
\text{Length of } EF = |y_2 - y_1| = |5 - (-1)| = |6| = 6
\]

Step 3: Length of \( FG \)

\( FG \) is a slant line. Use the distance formula: \( d = \sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2} \)
\[
\text{Length of } FG = \sqrt{(4 - (-2))^2 + (3 - 5)^2} = \sqrt{(6)^2 + (-2)^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10} \approx 6.32
\]

Step 4: Length of \( GH \)

\( GH \) is a vertical line (same \( x \)-coordinate). The length is the difference in \( y \)-coordinates.
\[
\text{Length of } GH = |y_4 - y_3| = |-1 - 3| = |-4| = 4
\]

Step 5: Length of \( EH \)

\( EH \) is a horizontal line (same \( y \)-coordinate). The length is the difference in \( x \)-coordinates.
\[
\text{Length of } EH = |x_4 - x_1| = |4 - (-2)| = |6| = 6
\]

Step 6: Perimeter of \( EFGH \)

Perimeter is the sum of all side lengths:
\[
\text{Perimeter} = EF + FG + GH + EH = 6 + 2\sqrt{10} + 4 + 6 = 16 + 2\sqrt{10} \approx 16 + 6.32 = 22.32
\]

(Note: The actual coordinates may vary based on the grid. Adjust the coordinates according to the given graph for precise calculation.)

Final Answers (Based on Assumed Coordinates):

• Length of \( EF \): \( \boldsymbol{6} \)
• Length of \( FG \): \( \boldsymbol{2\sqrt{10} \approx 6.32} \)
• Length of \( GH \): \( \boldsymbol{4} \)
• Length of \( EH \): \( \boldsymbol{6} \)
• Perimeter of \( EFGH \): \( \boldsymbol{16 + 2\sqrt{10} \approx 22.32} \)

Answer:

To solve the problem of finding the lengths of the sides and the perimeter of quadrilateral \( EFGH \), we need to assume the coordinates of the vertices (since the graph is provided, we'll infer typical grid coordinates). Let's assume the coordinates (for a standard grid) as follows (we'll define them to proceed):

Step 1: Define Coordinates (Assumed from Grid)

Let’s assume the coordinates of the vertices (estimating from the grid):

• \( E(x_1, y_1) \)
• \( F(x_2, y_2) \)
• \( G(x_3, y_3) \)
• \( H(x_4, y_4) \)

For example, if we assume:

• \( E(-2, -1) \)
• \( F(-2, 5) \)
• \( G(4, 3) \)
• \( H(4, -1) \)

Step 2: Length of \( EF \)

\( EF \) is a vertical line (same \( x \)-coordinate). The length is the difference in \( y \)-coordinates.
\[
\text{Length of } EF = |y_2 - y_1| = |5 - (-1)| = |6| = 6
\]

Step 3: Length of \( FG \)

\( FG \) is a slant line. Use the distance formula: \( d = \sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2} \)
\[
\text{Length of } FG = \sqrt{(4 - (-2))^2 + (3 - 5)^2} = \sqrt{(6)^2 + (-2)^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10} \approx 6.32
\]

Step 4: Length of \( GH \)

\( GH \) is a vertical line (same \( x \)-coordinate). The length is the difference in \( y \)-coordinates.
\[
\text{Length of } GH = |y_4 - y_3| = |-1 - 3| = |-4| = 4
\]

Step 5: Length of \( EH \)

\( EH \) is a horizontal line (same \( y \)-coordinate). The length is the difference in \( x \)-coordinates.
\[
\text{Length of } EH = |x_4 - x_1| = |4 - (-2)| = |6| = 6
\]

Step 6: Perimeter of \( EFGH \)

Perimeter is the sum of all side lengths:
\[
\text{Perimeter} = EF + FG + GH + EH = 6 + 2\sqrt{10} + 4 + 6 = 16 + 2\sqrt{10} \approx 16 + 6.32 = 22.32
\]

(Note: The actual coordinates may vary based on the grid. Adjust the coordinates according to the given graph for precise calculation.)

Final Answers (Based on Assumed Coordinates):

• Length of \( EF \): \( \boldsymbol{6} \)
• Length of \( FG \): \( \boldsymbol{2\sqrt{10} \approx 6.32} \)
• Length of \( GH \): \( \boldsymbol{4} \)
• Length of \( EH \): \( \boldsymbol{6} \)
• Perimeter of \( EFGH \): \( \boldsymbol{16 + 2\sqrt{10} \approx 22.32} \)