What is the perimeter of a triangle with corners at #(7 ,5 )#, #(8 ,2 )#, and #(4 ,7 )#?

Question

What is the perimeter of a triangle with corners at #(7 ,5  )#, #(8   ,2  )#, and #(4  ,7  )#?

Violet Aldrich · Accepted Answer

Perimeter of the triangle = #color(blue)(13.171# Units

Given:

The three vertices of a triangle is given to us.

Let the vertices be A, B, and C.

Hence, we have #color(blue)(A(7, 5), B(8, 2) and C(4, 7)# as shown below:

Observe that we have three segments: #bar(AB), bar(BC) and bar(AC)#

Find the magnitude of #bar(AB)# using the Distance formula:

Distance formula: #AB=sqrt[(x_2-x_1)^2+(y_2-y_1)^2)#

#color(red)("Step 1"#

Find the distance between the points:-

#color(blue)(A(7, 5) and B(8, 2)#

Let us say the points are #A(x_1, y_1) and B(x_2, y_2)#

Then, #x_1 = 7; x_2=8, y_1 = 5 and y_2=2#

Using the distance formula, we get

#sqrt[(8-7)^2+(2-5)^2]#

#rArr sqrt(1+9)#

#rArr sqrt(10)#

#~~ 3.16228#

Hence, #color(green)(bar(AB)~~3.16228#

#color(red)("Step 2"#

Find the distance between the points:-

#color(blue)(B(8, 2) and C(4, 7)#

Let us say the points are #B(x_1, y_1) and C(x_2, y_2)#

Then, #x_1 = 8; x_2=4, y_1 = 2 and y_2=7#

Using the distance formula, we get

#sqrt[(4-8)^2+(7-2)^2]#

#rArr sqrt(16+25)#

#rArr sqrt(41)#

#~~6.40312#

Hence, #color(green)(bar(BC)~~6.40312#

#color(red)("Step 3"#

Find the distance between the points:-

#color(blue)(A(7, 5) and C(4, 7)#

Let us say the points are #A(x_1, y_1) and C(x_2, y_2)#

Then, #x_1 = 7; x_2=4, y_1 = 5 and y_2=7#

Using the distance formula, we get

#sqrt[(4-7)^2+(7-5)^2]#

#rArr sqrt(9+4)#

#rArr sqrt(13)#

#~~ 3.60555#

Hence, #color(green)(bar(AC)~~3.60555#

#color(red)("Step 4"#

Perimeter of the triangle# = AB + BC + AC#

Perimeter #~~3.16228+6.40312+3.60555~~13.171#

Hence,

Perimeter of the triangle ABC #color(blue)(~~13.171# units.

View the image below for an illustration:

Ella Armstrong · Answer

undefined To find the perimeter of the triangle with coordinates (7, 5), (8, 2), and (4, 7), you need to calculate the distance between each pair of consecutive points and then sum up these distances.

Using the distance formula:

$ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $

Distance between (7, 5) and (8, 2):
$ \sqrt{(8 - 7)^2 + (2 - 5)^2} = \sqrt{(1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} $

Distance between (8, 2) and (4, 7):
$ \sqrt{(4 - 8)^2 + (7 - 2)^2} = \sqrt{(-4)^2 + (5)^2} = \sqrt{16 + 25} = \sqrt{41} $

Distance between (4, 7) and (7, 5):
$ \sqrt{(7 - 4)^2 + (5 - 7)^2} = \sqrt{(3)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} $

Now, add these distances together to get the perimeter:

$ \text{Perimeter} = \sqrt{10} + \sqrt{41} + \sqrt{13} $