Distance between two points

To find the distance between two points (in a 2D plane) we can use the pythagorean theorem:

a^2 + b^2 = c^2

Formula for distance between two points

We can derive this, however, by considering a to be the difference in the x-coordinates of the two points, b to be the difference in the y-coordinates of the two points, and c to be the distance between the two points:

(x_2 - x_1)^2 + (y_2 - y_1)^2 = d^2

Rearranging this to make d the subject gives us the formula for the distance between two points:

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Find the distance between the points (3, 4) and (7, 1).

• Values we know:
  • x_1 = 3
  • y_1 = 4
  • x_2 = 7
  • y_2 = 1
• Substitute into the formula:
  • d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
  • d = \sqrt{(7 - 3)^2 + (1 - 4)^2}
  • = \sqrt{(4)^2 + (-3)^2}
  • = \sqrt{16 + 9}
  • = \sqrt{25}
  • = 5
• Answer: The distance between the points (3, 4) and (7, 1) is 5 units.

Find the distance between the points (-2, 5) and (4, -1).

• Values we know:
  • x_1 = -2
  • y_1 = 5
  • x_2 = 4
  • y_2 = -1
• Substitute into the formula:
  • d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
  • d = \sqrt{(4 - (-2))^2 + (-1 - 5)^2}
  • = \sqrt{(6)^2 + (-6)^2}
  • = \sqrt{36 + 36}
  • = \sqrt{72}
  • = 6\sqrt{2}
• Answer: The distance between the points (-2, 5) and (4, -1) is 6\sqrt{2} units.