William is getting ready to enter his quilt into a competition at the state fair. Before he enters the quilt, he needs to stitch the top of the quilt to the backing. He created a map of the quilt with the coordinates A(1,4)
, B(1,−1)
, C(−3,−1)
, and D(−1,4)
. Each unit on the coordinate plane represents 1 foot. Find the perimeter of the quilt in order to determine how much thread William will need to stitch the quilt to the backing.
To find the perimeter of the quilt, we need to find the distance between each pair of consecutive points and then add them up.
The distance between points A and B can be found using the distance formula:
AB = √( (1-1)^2 + (-1-4)^2 ) = √(0 + 25) = √25 = 5
The distance between points B and C can be found using the distance formula:
BC = √( (-3-1)^2 + (-1-(-1))^2 ) = √(16 + 0) = √16 = 4
The distance between points C and D can be found using the distance formula:
CD = √( (-1-(-3))^2 + (4-(-1))^2 ) = √(4^2 + 5^2) = √(16 + 25) = √41
The distance between points D and A can be found using the distance formula:
DA = √( (1-(-1))^2 + (4-4)^2 ) = √(2^2 + 0) = √4 = 2
Now, we can add up the distances to find the perimeter:
Perimeter = AB + BC + CD + DA
Perimeter = 5 + 4 + √41 + 2
Perimeter ≈ 11 + √41 feet
Therefore, the perimeter of the quilt is approximately 11 + √41 feet. William will need this much thread to stitch the quilt to the backing.