Distance Formula
Apply the distance formula in Grade 10 algebra. Calculate d=√((x₂-x₁)²+(y₂-y₁)²) to find the exact distance between any two points on a coordinate plane.
Key Concepts
Property The distance $d$ between any two points with coordinates $(x 1, y 1)$ and $(x 2, y 2)$ is $$d = \sqrt{(x 2 x 1)^2 + (y 2 y 1)^2}.$$.
Find the distance between $(2, 3)$ and $(10, 9)$: $d = \sqrt{(10 2)^2 + (9 3)^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10$. Find the distance between $( 1, 7)$ and $(4, 5)$: $d = \sqrt{(4 ( 1))^2 + ( 5 7)^2} = \sqrt{5^2 + ( 12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13$.
This formula looks fancy, but it's just the Pythagorean theorem dressed up for a coordinate plane party. It finds the direct, straight line distance between any two points. Imagine drawing a right triangle connecting the points; the horizontal and vertical distances are the legs. The formula just calculates the hypotenuse, which is the distance you actually wanted to find.
Common Questions
What is the distance formula?
d = √((x₂-x₁)² + (y₂-y₁)²). It gives the straight-line distance between two points (x₁,y₁) and (x₂,y₂), derived from the Pythagorean theorem.
How do you use the distance formula for points (1,2) and (4,6)?
d = √((4-1)² + (6-2)²) = √(9 + 16) = √25 = 5. The distance between the points is exactly 5 units.
How is the distance formula related to the Pythagorean theorem?
The horizontal and vertical distances (x₂-x₁) and (y₂-y₁) form the two legs of a right triangle. The distance d is the hypotenuse: a² + b² = c² becomes d = √((Δx)² + (Δy)²).