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Bot GPT-4o To find the perimeter of the polygon with vertices at \((1, 3)\), \((7, 3)\), \((7, 7)\), and \((4, 7)\), we need to calculate the lengths of each of its sides and then add them together.
1. Calculate the distance between \((1, 3)\) and \((7, 3)\):
\[
\text{Distance} = \sqrt{(7 - 1)^2 + (3 - 3)^2} = \sqrt{6^2 + 0^2} = 6
\]
2. Calculate the distance between \((7, 3)\) and \((7, 7)\):
\[
\text{Distance} = \sqrt{(7 - 7)^2 + (7 - 3)^2} = \sqrt{0^2 + 4^2} = 4
\]
3. Calculate the distance between \((7, 7)\) and \((4, 7)\):
\[
\text{Distance} = \sqrt{(4 - 7)^2 + (7 - 7)^2} = \sqrt{(-3)^2 + 0^2} = 3
\]
4. Calculate the distance between \((4, 7)\) and \((1, 3)\):
\[
\text{Distance} = \sqrt{(1 - 4)^2 + (3 - 7)^2} = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = 5
\]
Now sum all the side lengths:
\[
6 + 4 + 3 + 5 = 18
\]
Therefore, the perimeter of the polygon is \(\boxed{18}\) units.