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) = √(0^2 + (-5)^2) = √25 = 5 feet
Similarly, the distance between points B and C is:
BC = √((1+3)^2 + (-1+1)^2) = √(4^2 + 0^2) = 4 feet
And the distance between points C and D is:
CD = √((-3+1)^2 + (-1-4)^2) = √((-2)^2 + (-5)^2) = √(4 + 25) = √29 feet
Finally, the distance between points D and A is:
DA = √((-1-1)^2 + (4-4)^2) = √((-2)^2 + 0^2) = √4 = 2 feet
Now, we add up all the distances to find the perimeter of the quilt:
Perimeter = AB + BC + CD + DA
Perimeter = 5 + 4 + √29 + 2
Perimeter = 11 + √29 feet
Therefore, the perimeter of the quilt is 11 + √29 feet. This is the amount of thread William will need to stitch the quilt to the backing.