Distance Formula Class 10 Maths: Examples & Questions

The distance formula is a fundamental concept in coordinate geometry used to find the straight-line distance between two points on a plane. It is based on the Pythagorean theorem and allows us to measure how far apart two points are, even if they are not aligned horizontally or vertically.

In this chapter, you will learn how to find the distance between two points whose coordinates are given.

Table of Contents

• What is the Distance Formula
• Applications of the Distance Formula
• Solved Examples on Distance Formula
  
  

What is the Distance Formula

The distance formula is used to find the straight-line distance between two points in a coordinate plane. If two points are P(x₁, y₁) and Q(x₂, y₂), then:

PQ = √((x₂ − x₁)² + (y₂ − y₁)²)

Let P(x₁, y₁) and Q(x₂, y₂) be any two points.

Then:

• PQ² = (x₂ − x₁)² + (y₂ − y₁)²
• PQ = √((x₂ − x₁)² + (y₂ − y₁)²)

This is the distance formula.

NOTE: Distance from origin O(0,0): OP = √(x² + y²)

Applications of the Distance Formula

• Finding the length of a line segment
• Checking collinearity (if AB + BC = AC)
• Determining type of triangle
• Finding perimeter of figures
• Used in maps, navigation, and construction
  
  

Solved Examples on Distance Formula

Example 1: Do A(1,2), B(4,6), C(7,2) form a triangle?

Solution:

AB = √((4 − 1)² + (6 − 2)²) = √(3² + 4²) = 5

BC = √((7 − 4)² + (2 − 6)²) = 5

CA = √((7 − 1)² + (2 − 2)²) = 6

Since two sides are equal, the triangle is isosceles.

Example 2: Check if P(1,2), Q(3,6), R(5,10) are collinear.

Solution:

PQ = 2√5

QR = 2√5

PR = 4√5

Since PQ + QR = PR, the points are collinear.

Example 3: Find relation between x and y if (x, y) is equidistant from (7,1) and (3,5).

Solution:

(x − 7)² + (y − 1)² = (x − 3)² + (y − 5)²

Simplifying:

x − y = 2