What is the perimeter of a triangle with corners at #(9 ,2 )#, #(6 ,3 )#, and #(4 ,1 )#?

Question

What is the perimeter of a triangle with corners at #(9  ,2  )#, #(6  ,3  )#, and #(4  ,1   )#?

Theodore Abad · Accepted Answer

See a solution process below:

To find the perimeter of the triangle formed by these three points we need to find the distance between each pair of points. 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 (9, 2) and (6, 3) #d = sqrt((color(red)(6) - color(blue)(9))^2 + (color(red)(3) - color(blue)(2))^2)# #d = sqrt(-3^2 + 1^2)# #d = sqrt(9 + 1)# #d = sqrt(2)# Distance Between (9, 2) and (4, 1) #d = sqrt((color(red)(4) - color(blue)(9))^2 + (color(red)(1) - color(blue)(2))^2)# #d = sqrt(-5^2 + (-1)^2)# #d = sqrt(25 + 1)# #d = sqrt(26)# Distance Between (6, 3) and (4, 1) #d = sqrt((color(red)(4) - color(blue)(6))^2 + (color(red)(1) - color(blue)(3))^2)# #d = sqrt(-2^2 + (-2)^2)# #d = sqrt(4 + 4)# #d = sqrt(8)# Therefore the perimeter of the triangle is: #p = sqrt(2) + sqrt(26) + sqrt(8)# We can factor out a #sqrt(2)# if desired: #p = sqrt(2) + sqrt(2 * 13) + sqrt(2 * 4)# #p = sqrt(2) + sqrt(2)sqrt(13) + sqrt(2)sqrt(4)# #p = 1sqrt(2) + sqrt(2)sqrt(13) + sqrt(2)sqrt(4)# #p =(1 + sqrt(13) + sqrt(4))sqrt(2)# #p =(1 + sqrt(13) + 2)sqrt(2)# #p =(3 + sqrt(13))sqrt(2)# Or #p = 3sqrt(2) + sqrt(13)sqrt(2)# #p = 3sqrt(2) + sqrt(13 * 2)# #p = 3sqrt(2) + sqrt(26)#

Sadie Allen · Answer

undefined To find the perimeter of a triangle, we need to calculate the sum of the lengths of its three sides using the distance formula.

Let's denote the coordinates of the vertices as follows:
$ A (x_1, y_1) = (9, 2) $
$ B (x_2, y_2) = (6, 3) $
$ C (x_3, y_3) = (4, 1) $

The distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:
$$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

Now, we'll calculate the distances between the vertices:
1. Distance between A and B:
$$ AB = \sqrt{(6 - 9)^2 + (3 - 2)^2} = \sqrt{(-3)^2 + (1)^2} = \sqrt{9 + 1} = \sqrt{10} $$
2. Distance between B and C:
$$ BC = \sqrt{(4 - 6)^2 + (1 - 3)^2} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} $$
3. Distance between C and A:
$$ CA = \sqrt{(9 - 4)^2 + (2 - 1)^2} = \sqrt{(5)^2 + (1)^2} = \sqrt{25 + 1} = \sqrt{26} $$

Now, we'll sum up these distances to find the perimeter of the triangle:
$$ Perimeter = AB + BC + CA = \sqrt{10} + \sqrt{8} + \sqrt{26} $$

This is the exact expression for the perimeter of the triangle, and if needed, you can calculate its approximate numerical value by evaluating the square roots.