What is the perimeter of a triangle with corners at #(5 ,2 )#, #(9 ,7 )#, and #(1 ,4 )#?

Question

What is the perimeter of a triangle with corners at #(5 ,2  )#, #(9 ,7  )#, and #(1  ,4 )#?

Aaliyah Anderson · Accepted Answer

The perimeter is 19.419, or #(sqrt(41)+sqrt(73)+2sqrt(5))#

Let's call each of the corners/vertices #A#, #B#, and #C#.

#A=(5,2)# #B=(9,7)# #C=(1,4)#

The Perimeter #P#, can be described as:

#P=bar(AB)+bar(AC)+bar(BC)#

where the side length of the triangle is represented by the value with the bar over it.

We can calculate the individual lengths by treating the #x# and #y# displacements as right triangles, and using The Pythagorean Theorem to calculate the length:

If #c=sqrt(a^2+b^2)#, and:

#a=Deltax=(x_2-x_1)# #b=Deltay=(y_2-y_1)# #c="side length"#

A general equation of any length can be rewritten as follows:

#c=sqrt((x_2-x_1)^2+(y_2-y_1)^2)#

Now, let's do the math for one leg, #bar(AB)#:

#bar(AB)=sqrt((x_B-x_A)^2+(y_B-y_A)^2)#

#bar(AB)=sqrt((9-5)^2+(7-2)^2)=sqrt(4^2+5^2)#

#bar(AB)=sqrt(16+25)#

#bar(AB)=sqrt(41)=6.403#

Repeating the process for #bar(BC)# and #bar(AC)#:

#bar(BC)=sqrt(73)=8.544#

#bar(AC)=sqrt(20)=2sqrt(5)=4.472#

Now we can calculate the perimeter, #P#:

#P=sqrt(41)+sqrt(73)+2sqrt(5)#

#P=6.403+8.544+4.472#

#color(green)(P=19.419#

Savannah Aguilar · Answer

undefined To find the perimeter of the triangle, you need to calculate the distance between each pair of points and then sum them up. You can use the distance formula $ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $ to find the distance between two points in a coordinate plane. Calculate the distances between the given points and then sum them up to find the perimeter.