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.

Answer 1

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.