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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(−3,4) . 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.(1 point)

1 answer

To find the perimeter of the quilt, we need to calculate the lengths of the sides formed by the points A(1,4), B(1,-1), C(-3,-1), and D(-3,4).

  1. Calculate the length of side AB:

    • The coordinates of A are (1, 4) and B are (1, -1).
    • The length can be calculated using the distance formula for vertical segments: \[ AB = |y_2 - y_1| = |-1 - 4| = |-5| = 5 \text{ feet} \]

  2. Calculate the length of side BC:

    • The coordinates of B are (1, -1) and C are (-3, -1).
    • The length can be calculated using the distance formula for horizontal segments: \[ BC = |x_2 - x_1| = |-3 - 1| = |-4| = 4 \text{ feet} \]

  3. Calculate the length of side CD:

    • The coordinates of C are (-3, -1) and D are (-3, 4).
    • The length can be calculated using the distance formula for vertical segments: \[ CD = |y_2 - y_1| = |4 - (-1)| = |4 + 1| = |5| = 5 \text{ feet} \]

  4. Calculate the length of side DA:

    • The coordinates of D are (-3, 4) and A are (1, 4).
    • The length can be calculated using the distance formula for horizontal segments: \[ DA = |x_2 - x_1| = |1 - (-3)| = |1 + 3| = |4| = 4 \text{ feet} \]

Now that we have the lengths of all four sides, we can calculate the perimeter \( P \) of the quilt: \[ P = AB + BC + CD + DA = 5 + 4 + 5 + 4 = 18 \text{ feet} \]

Thus, the perimeter of the quilt is \( \boxed{18} \) feet.