The problem asks you to graph five points $A(-2, -3), B(-6, 5), C(11, 5), D(7, 2), E(7, -3)$ and connect them in the given order to form a shape. Then, you need to find the area and perimeter of the shape.

Step 1: Plotting the Points

Here are the coordinates of the points:

• $A(-2, -3)$
• $B(-6, 5)$
• $C(11, 5)$
• $D(7, 2)$
• $E(7, -3)$

Step 2: Connect the points in the given order

The shape is formed by connecting these points in this order: A to B, B to C, C to D, D to E, and finally E back to A.

Step 3: Perimeter Calculation

The perimeter of the shape is the sum of the lengths of its sides. To calculate each side's length, we'll use the distance formula between two points:

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

Now, let's calculate the lengths of each side:

1. $AB = \sqrt{(-6 - (-2))^2 + (5 - (-3))^2} = \sqrt{(-4)^2 + (8)^2} = \sqrt{16 + 64} = \sqrt{80} \approx 8.94$
2. $BC = \sqrt{(11 - (-6))^2 + (5 - 5)^2} = \sqrt{17^2 + 0^2} = \sqrt{289} = 17$
3. $CD = \sqrt{(7 - 11)^2 + (2 - 5)^2} = \sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$
4. $DE = \sqrt{(7 - 7)^2 + (-3 - 2)^2} = \sqrt{0 + (-5)^2} = \sqrt{25} = 5$
5. $EA = \sqrt{(-2 - 7)^2 + (-3 - (-3))^2} = \sqrt{(-9)^2 + 0^2} = \sqrt{81} = 9$

So, the perimeter is: $P = AB + BC + CD + DE + EA = 8.94 + 17 + 5 + 5 + 9 = 44.94$

Step 4: Area Calculation

Since this is an irregular shape, we'll use the Shoelace Theorem (or the formula for the area of a polygon). The coordinates of the points in order are: $A(-2, -3), B(-6, 5), C(11, 5), D(7, 2), E(7, -3)$

Using the formula:

$\text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_4 + x_4y_5 + x_5y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_5 + y_5x_1) \right|$

Substitute the coordinates:

$\text{Area} = \frac{1}{2} \left| (-2)(5) + (-6)(5) + (11)(2) + (7)(-3) + (7)(-3) - ((-3)(-6) + 5(11) + 5(7) + 2(7) + (-3)(-2)) \right|$ $= \frac{1}{2} \left| -10 - 30 + 22 - 21 - 21 - (18 + 55 + 35 + 14 + 6) \right|$ $= \frac{1}{2} \left| -60 - (128) \right| = \frac{1}{2} \times 188 = 94$

Thus, the area of the shape is 94 square units.

Final Answer:

• Perimeter = 44.94 units
• Area = 94 square units

Would you like any further details or explanations? Let me know if you have any questions!

Here are some additional related questions:

1. What is the Shoelace Theorem, and how is it derived?
2. Can the distance formula be generalized to higher dimensions?
3. What are some other methods for calculating the area of irregular polygons?
4. How would the calculations change if the shape was not closed (i.e., missing a side)?
5. How can graphing software be used to verify the area and perimeter?

Tip: When working with multiple points and finding distances or areas, keeping track of your work step-by-step helps reduce errors in calculation.