Question

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

Answer 1

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

Explanation:

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: