To find the perimeter of a triangle with vertices A(2, 5), B(2, -2), and C(5, -2), we need to follow these steps:
### Step 1: Calculate the Length of Each Side
We use the distance formula:
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Length of side AB (between A and B):
- A = (2, 5)
- B = (2, -2)
[tex]\[ AB = \sqrt{(2 - 2)^2 + (-2 - 5)^2} \][/tex]
[tex]\[ AB = \sqrt{0 + (-7)^2} \][/tex]
[tex]\[ AB = \sqrt{49} \][/tex]
[tex]\[ AB = 7.0 \][/tex]
Length of side BC (between B and C):
- B = (2, -2)
- C = (5, -2)
[tex]\[ BC = \sqrt{(5 - 2)^2 + (-2 - (-2))^2} \][/tex]
[tex]\[ BC = \sqrt{3^2 + 0} \][/tex]
[tex]\[ BC = \sqrt{9} \][/tex]
[tex]\[ BC = 3.0 \][/tex]
Length of side CA (between C and A):
- C = (5, -2)
- A = (2, 5)
[tex]\[ CA = \sqrt{(2 - 5)^2 + (5 - (-2))^2} \][/tex]
[tex]\[ CA = \sqrt{(-3)^2 + 7^2} \][/tex]
[tex]\[ CA = \sqrt{9 + 49} \][/tex]
[tex]\[ CA = \sqrt{58} \][/tex]
[tex]\[ CA \approx 7.6 \][/tex]
### Step 2: Calculate the Perimeter
The perimeter of the triangle is the sum of the lengths of its sides:
[tex]\[ \text{Perimeter} = AB + BC + CA \][/tex]
[tex]\[ \text{Perimeter} = 7.0 + 3.0 + 7.6 \][/tex]
[tex]\[ \text{Perimeter} = 17.6 \][/tex]
### Step 3: Round the Perimeter
Since the calculated perimeter already ends in one decimal place, no further rounding is necessary.
Thus, the perimeter of the triangle is [tex]\(\boxed{17.6}\)[/tex].
