Step 1: Find the length of segment $ST$

The length of the segment $ST$ can be calculated using the distance formula, which is:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

where the coordinates of $S(1, -1)$ and $T(7, 7)$ are given. Plugging the values into the distance formula:

$d = \sqrt{(7 - 1)^2 + (7 - (-1))^2}$ $d = \sqrt{6^2 + 8^2}$ $d = \sqrt{36 + 64}$ $d = \sqrt{100}$ $d = 10$

So, the length of the side $ST$ is 10 units.

Step 2: Find the perimeter of the rectangle

The perimeter of a rectangle is given by:

$P = 2l + 2w$

where $l$ is the length and $w$ is the width. We are told that the perimeter is 36 units, so:

$36 = 2(10) + 2w$ $36 = 20 + 2w$ $2w = 36 - 20$ $2w = 16$ $w = 8$

So, the width of the rectangle is 8 units.

Step 3: Find the area of the rectangle

The area $A$ of a rectangle is given by:

$A = l \times w$

Substituting the length and width:

$A = 10 \times 8 = 80$

Final Answer:

The area of the rectangle is $\boxed{80}$ square units.

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Would you like further details on any of the steps or additional questions on similar topics? Here are some questions to consider:

1. How would you calculate the area if the rectangle were oriented differently?
2. What happens to the area if the perimeter changes to 40 units while keeping one side fixed?
3. How can you use the coordinates to determine the direction in which the rectangle is oriented?
4. How do you find the coordinates of the other two vertices of the rectangle?
5. How do you check if the rectangle is aligned with the axes?

Tip: When solving geometry problems, visualizing the coordinates and using the distance formula to calculate lengths and diagonals can make it easier to apply other formulas like perimeter and area.