What is the perimeter of a triangle with corners at #(2 ,6 )#, #(4 ,5 )#, and #(3 ,1 )#?

Question

What is the perimeter of a triangle with corners at #(2  ,6  )#, #(4 ,5  )#, and #(3 ,1  )#?

Claire Akin · Accepted Answer

See a solution process below:

The perimeter of a triangle is: #p = s_1 + s_2 + s_3# Where: #s_1#, #s_2# and #s_3# are the lengths of the sides of the triangle. We need to find the distance between each of the points in the problem. 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)# Distance Between (2, 6) and (4, 5) is: #s_1 = sqrt((color(red)(4) - color(blue)(2))^2 + (color(red)(5) - color(blue)(6))^2)# #s_1 = sqrt(2^2 + (-1)^2)# #s_1 = sqrt(4 + 1)# #s_1 = sqrt(5)# Distance Between (2, 6) and (3, 1) is: #s_2 = sqrt((color(red)(3) - color(blue)(2))^2 + (color(red)(1) - color(blue)(6))^2)# #s_2 = sqrt(1^2 + (-5)^2)# #s_2 = sqrt(1 + 25)# #s_2 = sqrt(26)# Distance Between (4, 5) and (3, 1) is: #s_3 = sqrt((color(red)(3) - color(blue)(4))^2 + (color(red)(1) - color(blue)(5))^2)# #s_3 = sqrt((-1)^2 + (-4)^2)# #s_3 = sqrt(1 + 16)# #s_3 = sqrt(17)# The Perimeter of the Triangle is: #p = s_1 + s_2 + s_3# #p = sqrt(5) + sqrt(26) + sqrt(17)# If you need a single number answer: #p = 11.458# rounded to the nearest thousandth.

Wyatt Anderson · Answer

undefined To find the perimeter of the triangle with corners at (2, 6), (4, 5), and (3, 1), we need to find the lengths of each side and then sum them up.

Using the distance formula, the lengths of the sides are:

Side 1: between (2, 6) and (4, 5)
$$ \sqrt{(4-2)^2 + (5-6)^2} = \sqrt{2^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5} $$

Side 2: between (2, 6) and (3, 1)
$$ \sqrt{(3-2)^2 + (1-6)^2} = \sqrt{1^2 + (-5)^2} = \sqrt{1 + 25} = \sqrt{26} $$

Side 3: between (4, 5) and (3, 1)
$$ \sqrt{(3-4)^2 + (1-5)^2} = \sqrt{(-1)^2 + (-4)^2} = \sqrt{1 + 16} = \sqrt{17} $$

Now, sum up the lengths of the sides to find the perimeter:

$$ \text{Perimeter} = \sqrt{5} + \sqrt{26} + \sqrt{17} $$