Asquare park measures 170 feet along each side. two paved paths run from each corner to the opposite corner and extend 3 feet inwards from each corner, as shown. what is the total area, in square feet, taken by the paths?

The total area, in square feet, taken by the paths is 2,004

Step-by-step explanation:

see the attached figure with lines to better understand the problem

I can divide the figure into four right  triangles, one small square and four rectangles

step 1

Find the area of the right triangle of each corner of the path

The area of the triangle is

substitute the given values

step 2

Find the hypotenuse of the right triangle

Applying Pythagoras Theorem

Let

d -----> hypotenuse of the right triangle

simplify

The hypotenuse of the right triangle is equal to the width of the path

step 2

Find the area of the small square of the path

The area is

we have

 ----> the width of the path

substitute

step 3

Find the length of the diagonal of the square park

Applying Pythagoras Theorem

Let

D -----> diagonal of the square park

simplify

step 4

Find the height of each right triangle on each corner

The height will be equal to the width of the path divided by two, because is a 45-90-45 right triangle

step 5

Find the area of each rectangle of the path

The area of rectangle is

we have

----> width of the path

Find the length of each rectangle of the path

where

D is the diagonal of the park

h is the height of the right triangle in the corner

d is the width of the path (length side of the small square of the path)

substitute the values

Find the area of each rectangle of the path

we have

substitute

step 6

Find the area of the paths

Remember

The total area of the paths is equal to the area of four right  triangles, one small square and four rectangles

so

substitute

therefore

The total area, in square feet, taken by the paths is 2,004