find the perimeter of the polygon with vertices A(-5,-3) B(1,0) C (3,-4) D(-3,-7)

The distance between two given points P1(x1,y1) and P2(x2,y2) in the coordinate plane is given by the expression:

distance P1-P2 = sqrt((x2-x1)^2+(y2-y1)^2)
A(-5,-3) B(1,0) C (3,-4) D(-3,-7)
For example, the distance
AB = sqrt((1-(-5))^2+(0-(-3))^2)
= sqrt(6^2+3^2)
= sqrt(45)
= 6.708...(approx.)

Proceed the same way for BC, CD and CA and add up the four distances to get the perimeter.

To find the perimeter of a polygon, you need to calculate the sum of the lengths of all its sides. Each side's length can be found using the distance formula. Let's go step by step to find the perimeter of the given polygon with vertices A(-5,-3), B(1,0), C(3,-4), and D(-3,-7).

Step 1: Calculate the distance between vertices A and B.
The distance formula between two points (x1, y1) and (x2, y2) is given by:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

In this case, (x1, y1) = (-5, -3) and (x2, y2) = (1, 0).
So the distance between A and B is:
distance_AB = sqrt((1 - (-5))^2 + (0 - (-3))^2)
= sqrt((6)^2 + (3)^2)
= sqrt(36 + 9)
= sqrt(45)
= 6.708

Step 2: Calculate the distance between vertices B and C.
Using the same formula with (x1, y1) = (1, 0) and (x2, y2) = (3, -4):
distance_BC = sqrt((3 - 1)^2 + (-4 - 0)^2)
= sqrt((2)^2 + (-4)^2)
= sqrt(4 + 16)
= sqrt(20)
= 4.472

Step 3: Calculate the distance between vertices C and D.
Using the same formula with (x1, y1) = (3, -4) and (x2, y2) = (-3, -7):
distance_CD = sqrt((-3 - 3)^2 + (-7 - (-4))^2)
= sqrt((-6)^2 + (-3)^2)
= sqrt(36 + 9)
= sqrt(45)
= 6.708

Step 4: Calculate the distance between vertices D and A.
Using the same formula with (x1, y1) = (-3, -7) and (x2, y2) = (-5, -3):
distance_DA = sqrt((-5 - (-3))^2 + (-3 - (-7))^2)
= sqrt((-2)^2 + (4)^2)
= sqrt(4 + 16)
= sqrt(20)
= 4.472

Step 5: Add up the distances of all the sides to find the perimeter.
perimeter = distance_AB + distance_BC + distance_CD + distance_DA
= 6.708 + 4.472 + 6.708 + 4.472
= 22.36

Therefore, the perimeter of the given polygon is 22.36 units.