Mathematics
Grade 10
15 min
Find the distance between two points
Find the distance between two points
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
State the distance formula from memory.
Correctly identify and label the coordinates (x₁, y₁) and (x₂, y₂) from two given points.
Calculate the distance between two points on the coordinate plane using the distance formula.
Explain the relationship between the distance formula and the Pythagorean theorem.
Apply the distance formula to find the lengths of the sides of a polygon given its vertices.
Use side lengths calculated with the distance formula to classify a triangle as scalene, isosceles, or equilateral.
Ever wondered how your phone's GPS calculates the 'as the crow flies' distance between you and your destination? 🗺️ It's all about using a powerful formula to find the distance between two points!
In this tutorial, you will lear...
2
Key Concepts & Vocabulary
TermDefinitionExample
Coordinate PlaneA two-dimensional plane formed by the intersection of a horizontal line called the x-axis and a vertical line called the y-axis.The familiar grid system where you plot points like (4, -2).
Ordered Pair (x, y)A pair of numbers used to locate a point on a coordinate plane. The first number (x-coordinate) indicates the horizontal position, and the second number (y-coordinate) indicates the vertical position.The point P(3, 5) is located 3 units to the right of the origin and 5 units up.
Line SegmentA part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints.The segment connecting point A(1, 2) and point B(5, 2), denoted as AB.
DistanceThe length of the straight line segment connecting two points....
3
Core Formulas
The Distance Formula
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
This formula calculates the distance 'd' between any two points (x₁, y₁) and (x₂, y₂). It is derived directly from the Pythagorean theorem by treating the line segment between the points as the hypotenuse of a right triangle.
Pythagorean Theorem (Geometric Basis)
a^2 + b^2 = c^2
This theorem is the foundation of the distance formula. The horizontal distance |x₂ - x₁| acts as side 'a', the vertical distance |y₂ - y₁| acts as side 'b', and the direct distance 'd' between the points acts as the hypotenuse 'c'.
4 more steps in this tutorial
Sign up free to access the complete tutorial with worked examples and practice.
Sign Up Free to ContinueSample Practice Questions
Challenging
The vertices of a triangle are J(1, 1), K(5, 4), and L(8, 0). How can you verify if it is a right triangle using the distance formula?
A.Check if JK = KL = JL.
B.Check if JK² + KL² = JL².
C.Check if JK + KL > JL.
D.Check if JK² + JL² = KL².
Challenging
A circle has its center at C(-2, 5). The point P(4, -3) lies on the circle. What is the length of the circle's diameter?
A.5
B.10
C.20
D.100
Challenging
Find the coordinates of a point on the y-axis that is equidistant from A(4, 4) and B(-3, 5).
A.(0, 1)
B.(0, -1)
C.(1, 0)
D.(0, 2)
Want to practice and check your answers?
Sign up to access all questions with instant feedback, explanations, and progress tracking.
Start Practicing Free