Finding Perimeter on a Coordinate Plane

📏 Understanding Perimeter on a Coordinate Plane

Finding the perimeter of a shape on a coordinate plane involves determining the length of each side and then summing those lengths. Here's a detailed approach:

Steps to Calculate Perimeter

1. Plot the Points: Plot all given coordinate points on the coordinate plane.
2. Connect the Points: Connect the points in the order they are given to form the shape.
3. Calculate Side Lengths: Use the distance formula or Pythagorean theorem to find the length of each side.
4. Sum the Lengths: Add up all the side lengths to find the perimeter.

📐 Distance Formula

The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

Example:

import math

def distance(x1, y1, x2, y2):
 return math.sqrt((x2 - x1)**2 + (y2 - y1)**2)

# Example points
x1, y1 = 1, 2
x2, y2 = 4, 6

d = distance(x1, y1, x2, y2)
print(f"The distance between ({x1}, {y1}) and ({x2}, {y2}) is: {d}")

📐 Pythagorean Theorem

If the sides of the shape are not parallel to the x or y-axis, you can form a right triangle and use the Pythagorean theorem ($a^2 + b^2 = c^2$) to find the length of the side.

Example:

import math

def pythagorean_distance(x1, y1, x2, y2):
 a = abs(x2 - x1)
 b = abs(y2 - y1)
 c = math.sqrt(a**2 + b**2)
 return c

# Example points
x1, y1 = 1, 2
x2, y2 = 4, 6

d = pythagorean_distance(x1, y1, x2, y2)
print(f"The distance between ({x1}, {y1}) and ({x2}, {y2}) using Pythagorean theorem is: {d}")

📝 Example Calculation

Consider a triangle with vertices A(1, 1), B(4, 5), and C(7, 1). Let's find its perimeter.

• AB: $d = \sqrt{(4-1)^2 + (5-1)^2} = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$
• BC: $d = \sqrt{(7-4)^2 + (1-5)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{25} = 5$
• CA: $d = \sqrt{(1-7)^2 + (1-1)^2} = \sqrt{(-6)^2 + 0^2} = \sqrt{36} = 6$

Perimeter = AB + BC + CA = 5 + 5 + 6 = 16 units.