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

Jan 28, 2018

$16.07$ units

#### Explanation:

To find the perimeter, we first need to find the lengths of the sides of the triangle.

First plot the coordinates of the points, and connect the points to form a triangle. This only needs to be a rough sketch, just so you can be clear where each point is in relation to each other.

We find the lengths of each side using the distance formula. If the end points of each line have coordinates color(blue)(( x_1 , y_1) and color(blue)((x_2,y_2), then the distance between each point is given by:

$d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{2}\right)}^{2}}$

This is the $\textcolor{b l u e}{\text{ Distance Formula}}$

We can see how this works by forming right angled triangles on each side of our given triangle. This is represented in the diagram below,

Each side of our given triangle is the hypotenuse of the right angled triangle formed on it.

By $\textcolor{b l u e}{\text{Pythagoras' Theorem}}$

The sum of the squares of the two sides equals the square on the hypotenuse. So the distance formula is just Pythagoras' theorem.

Now to our triangle.

We find each length separately:

We will start with the side represented in green on the diagram:

$d = \sqrt{{\left(7 - 1\right)}^{2} + {\left(5 - 2\right)}^{2}} = \sqrt{{6}^{2} + {3}^{2}} = \sqrt{44}$

Now the side represented in red:

$d = \sqrt{{\left(4 - 1\right)}^{2} + {\left(7 - 2\right)}^{2}} = \sqrt{{3}^{2} + {5}^{2}} = \sqrt{34}$

And the now side represented in blue:

$d = \sqrt{{\left(7 - 4\right)}^{2} + {\left(7 - 5\right)}^{2}} = \sqrt{{3}^{2} + {2}^{2}} = \sqrt{13}$

The perimeter of a triangle is the sum of the lengths of its sides:

$\therefore$

$\sqrt{44} + \sqrt{34} + \sqrt{13} = 16.07$ units. ( 2 .d.p.)

Summary

For finding perimeter

• plot the points and form a triangle.

• Find the lengths of each side using distance formula.

• Sum the lengths to find the perimeter.