To find the perimeter of the triangle we must find the length of each segment with the endpoints in the problem. Then we can add these lengths to find the perimeter of the triangle.
We need to find the length of segments:
d_1 is the distance between (1, 2) " and " (2, 3)
d_2 is the distance between (2, 3) " and " (4, 1)
d_3 is the distance between (4, 1) " and " (1, 2)
The formula for calculating the distance between two points is:
d = sqrt((color(red)(x_2) - color(blue)(x_1))^2 + (color(red)(y_2) - color(blue)(y_1))^2)
Length d_1:
d_1 = sqrt((color(red)(2) - color(blue)(1))^2 + (color(red)(3) - color(blue)(2))^2) = sqrt(1^2 + 1^2) = sqrt(1 + 1) = sqrt(2)
Length d_2:
d_2 = sqrt((color(red)(4) - color(blue)(2))^2 + (color(red)(1) - color(blue)(3))^2) = sqrt(2^2 + (-2)^2) = sqrt(4 + 4) = sqrt(8)
= sqrt(4 * 2) = sqrt(4)sqrt(2) = 2sqrt(2)
Length d_3:
d_3 = sqrt((color(red)(1) - color(blue)(4))^2 + (color(red)(2) - color(blue)(1))^2) = sqrt((-3)^2 + 1^2) = sqrt(9 + 1) = sqrt(10)
= sqrt(5 * 2) = sqrt(5)sqrt(2)
The perimeter p is:
p = d_1 + d_2 + d_3
Substituting gives:
p = sqrt(2) + 2sqrt(2) + sqrt(5)sqrt(2)
p = 1sqrt(2) + 2sqrt(2) + sqrt(5)sqrt(2)
p = (1 + 2 + sqrt(5))sqrt(2)
p = (3 + sqrt(5))sqrt(2)
Or
p = 7.405 rounded to the nearest thousandth.
