To find the perimeter of the triangle whose vertices are [tex]\((2, -3)\)[/tex], [tex]\((-7, 0)\)[/tex], and [tex]\((-3, -4)\)[/tex], we need to calculate the lengths of each side of the triangle and then sum these lengths.
### Step 1: Calculate the Length of Each Side
The distance between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] can be calculated using the distance formula:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
#### Side 1: Between [tex]\((2, -3)\)[/tex] and [tex]\((-7, 0)\)[/tex]
Use the distance formula:
[tex]\[
\text{Side 1} = \sqrt{((-7) - 2)^2 + (0 - (-3))^2}
= \sqrt{(-9)^2 + 3^2}
= \sqrt{81 + 9}
= \sqrt{90}
= 3\sqrt{10}
\][/tex]
#### Side 2: Between [tex]\((-7, 0)\)[/tex] and [tex]\((-3, -4)\)[/tex]
Use the distance formula:
[tex]\[
\text{Side 2} = \sqrt{((-3) - (-7))^2 + ((-4) - 0)^2}
= \sqrt{(4)^2 + (-4)^2}
= \sqrt{16 + 16}
= \sqrt{32}
= 4\sqrt{2}
\][/tex]
#### Side 3: Between [tex]\((-3, -4)\)[/tex] and [tex]\((2, -3)\)[/tex]
Use the distance formula:
[tex]\[
\text{Side 3} = \sqrt{(2 - (-3))^2 + (-3 - (-4))^2}
= \sqrt{(5)^2 + 1^2}
= \sqrt{25 + 1}
= \sqrt{26}
\][/tex]
### Step 2: Calculate the Perimeter
The perimeter [tex]\(P\)[/tex] of the triangle is the sum of the lengths of its sides:
[tex]\[
P = \text{Side 1} + \text{Side 2} + \text{Side 3}
\][/tex]
Substituting the calculated lengths:
[tex]\[
P = 3\sqrt{10} + 4\sqrt{2} + \sqrt{26}
\][/tex]
The exact perimeter of the triangle is:
[tex]\[
3\sqrt{10} + 4\sqrt{2} + \sqrt{26}
\][/tex]
So the perimeter of the triangle is:
[tex]\[
20.242706743590304
\][/tex]
This includes all steps and shows how to use the distance formula to find the lengths and sum them to find the perimeter.
