To find the perimeter of a shape using coordinates, you calculate the length of each side using the distance formula between consecutive vertices and then sum these lengths.
Understanding Perimeter and Coordinates
Perimeter is the total distance around the outside of a two-dimensional shape. When a shape is plotted on a coordinate plane, its vertices are given as ordered pairs $(x, y)$. To find the perimeter, you need a way to measure the distance between these points.
The Key Tool: The Distance Formula
The fundamental tool for calculating the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ in a coordinate plane is the distance formula:
$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
This formula is derived from the Pythagorean theorem.
Step-by-Step Guide to Finding Perimeter
The process for finding the perimeter of a polygon using coordinates is straightforward:
- Identify the vertices: List the coordinates of each vertex of the polygon in order as you move around the shape (e.g., $A(x_1, y_1)$, $B(x_2, y_2)$, $C(x_3, y_3)$, etc.).
- Calculate the length of each side: Use the distance formula to find the length of each segment connecting consecutive vertices.
- Find the distance between the first and second vertex.
- Find the distance between the second and third vertex.
- Continue this process for all consecutive pairs of vertices.
- Finally, find the distance between the last vertex and the first vertex (closing the shape).
- Based on the provided reference, use the coordinates of each vertex to find the length of each of the four sides through the distance formula. (This applies specifically to a quadrilateral, but the principle extends).
- Sum the side lengths: Add up the lengths of all the sides you calculated.
- The perimeter is given by the sum of the four sides. (Again, this applies specifically to a quadrilateral as per the reference, but the method is general for any polygon).
Example: Finding the Perimeter of a Quadrilateral
Let's find the perimeter of a quadrilateral with vertices A(1, 1), B(4, 1), C(4, 5), and D(1, 5).
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Vertices: A(1, 1), B(4, 1), C(4, 5), D(1, 5).
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Calculate Side Lengths:
- Side AB: Using A(1, 1) and B(4, 1)
$d_{AB} = \sqrt{(4 - 1)^2 + (1 - 1)^2} = \sqrt{3^2 + 0^2} = \sqrt{9} = 3$ - Side BC: Using B(4, 1) and C(4, 5)
$d_{BC} = \sqrt{(4 - 4)^2 + (5 - 1)^2} = \sqrt{0^2 + 4^2} = \sqrt{16} = 4$ - Side CD: Using C(4, 5) and D(1, 5)
$d_{CD} = \sqrt{(1 - 4)^2 + (5 - 5)^2} = \sqrt{(-3)^2 + 0^2} = \sqrt{9} = 3$ - Side DA: Using D(1, 5) and A(1, 1)
$d_{DA} = \sqrt{(1 - 1)^2 + (1 - 5)^2} = \sqrt{0^2 + (-4)^2} = \sqrt{16} = 4$
Side Vertices Calculation Length AB (1,1), (4,1) $\sqrt{(4-1)^2 + (1-1)^2}$ 3 BC (4,1), (4,5) $\sqrt{(4-4)^2 + (5-1)^2}$ 4 CD (4,5), (1,5) $\sqrt{(1-4)^2 + (5-5)^2}$ 3 DA (1,5), (1,1) $\sqrt{(1-1)^2 + (1-5)^2}$ 4 - Side AB: Using A(1, 1) and B(4, 1)
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Sum the Side Lengths:
Perimeter = $d{AB} + d{BC} + d{CD} + d{DA} = 3 + 4 + 3 + 4 = 14$
The perimeter of the quadrilateral is 14 units.
Applying to Other Polygons
This method using the distance formula applies to any polygon regardless of the number of sides (triangle, pentagon, hexagon, etc.). You simply find the distance between each adjacent pair of vertices around the shape and sum all those distances.
