Let's determine the perimeter of the triangle formed by the vertices [tex]\((-10, -4)\)[/tex], [tex]\((-1, 2)\)[/tex], and [tex]\((-4, 5)\)[/tex]. The perimeter of a triangle is the sum of the lengths of its three sides.
We will label the vertices as follows:
- [tex]\( A = (-10, -4) \)[/tex]
- [tex]\( B = (-1, 2) \)[/tex]
- [tex]\( C = (-4, 5) \)[/tex]
### Step 1: Calculate the Length of Side [tex]\( AB \)[/tex]
To find the length of the side [tex]\( AB \)[/tex] (the distance between points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]), we use the distance formula:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
For [tex]\( AB \)[/tex]:
- [tex]\( A = (-10, -4) \)[/tex]
- [tex]\( B = (-1, 2) \)[/tex]
Substituting into the distance formula:
[tex]\[ AB = \sqrt{(-1 - (-10))^2 + (2 - (-4))^2} \][/tex]
[tex]\[ AB = \sqrt{(-1 + 10)^2 + (2 + 4)^2} \][/tex]
[tex]\[ AB = \sqrt{(9)^2 + (6)^2} \][/tex]
[tex]\[ AB = \sqrt{81 + 36} \][/tex]
[tex]\[ AB = \sqrt{117} \][/tex]
[tex]\[ AB \approx 10.82 \][/tex]
### Step 2: Calculate the Length of Side [tex]\( BC \)[/tex]
For [tex]\( BC \)[/tex]:
- [tex]\( B = (-1, 2) \)[/tex]
- [tex]\( C = (-4, 5) \)[/tex]
Substituting into the distance formula:
[tex]\[ BC = \sqrt{(-4 - (-1))^2 + (5 - 2)^2} \][/tex]
[tex]\[ BC = \sqrt{(-4 + 1)^2 + (5 - 2)^2} \][/tex]
[tex]\[ BC = \sqrt{(-3)^2 + (3)^2} \][/tex]
[tex]\[ BC = \sqrt{9 + 9} \][/tex]
[tex]\[ BC = \sqrt{18} \][/tex]
[tex]\[ BC \approx 4.24 \][/tex]
### Step 3: Calculate the Length of Side [tex]\( CA \)[/tex]
For [tex]\( CA \)[/tex]:
- [tex]\( C = (-4, 5) \)[/tex]
- [tex]\( A = (-10, -4) \)[/tex]
Substituting into the distance formula:
[tex]\[ CA = \sqrt{(-10 - (-4))^2 + (-4 - 5)^2} \][/tex]
[tex]\[ CA = \sqrt{(-10 + 4)^2 + (-4 - 5)^2} \][/tex]
[tex]\[ CA = \sqrt{(-6)^2 + (-9)^2} \][/tex]
[tex]\[ CA = \sqrt{36 + 81} \][/tex]
[tex]\[ CA = \sqrt{117} \][/tex]
[tex]\[ CA \approx 10.82 \][/tex]
### Step 4: Calculate the Perimeter of the Triangle
The perimeter [tex]\( P \)[/tex] of the triangle is the sum of the lengths of the sides [tex]\( AB \)[/tex], [tex]\( BC \)[/tex], and [tex]\( CA \)[/tex]:
[tex]\[ P = AB + BC + CA \][/tex]
[tex]\[ P \approx 10.82 + 4.24 + 10.82 \][/tex]
[tex]\[ P \approx 25.88 \][/tex]
Therefore, the perimeter of the triangle is approximately [tex]\( 25.88 \)[/tex] units.
