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find the perimeter and area of this polygon

To find the perimeter and area of the triangle formed by the points \((-5, -4)\), \((2, -2)\), and \((1, 3)\), we can use the distance formula to calculate the lengths of the sides of the triangle and then use those lengths to find the perimeter and area.

### Step 1: Find the lengths of the sides using the distance formula

The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

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

Let's calculate the distances between each pair of points:

1. **Distance between \((-5, -4)\) and \((2, -2)\):**

\[
d_1 = \sqrt{(2 - (-5))^2 + (-2 - (-4))^2} = \sqrt{(2 + 5)^2 + (-2 + 4)^2} = \sqrt{7^2 + 2^2} = \sqrt{49 + 4} = \sqrt{53}
\]

2. **Distance between \((2, -2)\) and \((1, 3)\):**

\[
d_2 = \sqrt{(1 - 2)^2 + (3 - (-2))^2} = \sqrt{(-1)^2 + (3 + 2)^2} = \sqrt{1 + 25} = \sqrt{26}
\]

3. **Distance between \((1, 3)\) and \((-5, -4)\):**

\[
d_3 = \sqrt{(-5 - 1)^2 + (-4 - 3)^2} = \sqrt{(-6)^2 + (-7)^2} = \sqrt{36 + 49} = \sqrt{85}
\]

### Step 2: Calculate the perimeter of the triangle

The perimeter \(P\) of the triangle is the sum of the lengths of its sides:

\[
P = d_1 + d_2 + d_3 = \sqrt{53} + \sqrt{26} + \sqrt{85}
\]

### Step 3: Calculate the area of the triangle

We can use the **Shoelace formula** (or the formula for the area of a triangle given three vertices):

\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]

Plugging in the coordinates:

\[
\text{Area} = \frac{1}{2} \left| (-5)((-2) - 3) + 2(3 - (-4)) + 1((-4) - (-2)) \right|
\]

Simplify the expression:

\[
= \frac{1}{2} \left| (-5)(-5) + 2(7) + 1(-2) \right|
\]

\[
= \frac{1}{2} \left| 25 + 14 - 2 \right|
\]

\[
= \frac{1}{2} \left| 37 \right| = \frac{37}{2} = 18.5
\]

### Step 4: Final Results

- **Perimeter:** \( \sqrt{53} + \sqrt{26} + \sqrt{85} \approx 7.28 + 5.10 + 9.22 = 21.6 \)
- **Area:** \( 18.5 \) square units.

Would you like more details, or do you have any questions?

### Related Questions to Explore:
1. How do you find the distance between two points in 3D space?
2. What is the Shoelace formula used for, and can it be generalized to any polygon?
3. How would the calculations change if one point was moved to a different location?
4. How do you determine if three given points form a right triangle?
5. What are other methods to find the area of a triangle, besides the Shoelace formula?

#### Tip:
When calculating distances on a coordinate plane, always double-check your signs and ensure you square the differences correctly!

Find the perimeter and area of this polygon

Learn how to calculate the perimeter and area of a triangle with vertices at (-5, -4), (2, -2), and (1, 3) using the distance formula and Shoelace theorem. This guide covers geometry concepts and is suitable for high school students in Grades 9-12.

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