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Distance between Two Points Formula is used to calculate the distance between two points using the given coordinates. Distance between two points is the length of a line segment that connects two points. The distance between two points can be calculated in coordinate geometry by calculating the length of the line segment joining the two given coordinates. The points are defined by their coordinates along the X-axis and Y-axis. The coordinates of a point are a pair of values that depicts the exact location of a point. The Distance between Two Points Formula is given as d =√[(x2 – x1)2 + (y2 – y1)2]. The distance between two points formula is also an application of the Pythagoras theorem.
Read More: NCERT Solutions for Class 10 Maths Coordinate Geometry
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Key Terms: Distance between Two Points, Coordinates, Line Segment, Coordinate Plane, Pythagoras Theorem, Distance Between Two Points Formula
Distance Between Two Points
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The distance between two points is the length of a line segment that connects two points. The distance between two points can be found by calculating the length of the line segment connecting the given two points. The distance between the two points is always positive.
The video below explains this:
Coordinate Geometry Detailed Video Explanation:
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Distance Between Two Points Formula
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Distance Between Two Points Formula is used to calculate the distance between two points using the given coordinates. The distance between two points formula is usually given by d = √[(x2 – x1)2 + (y2 – y1)2]. The given formula is used to find the distance between any two points on a coordinate plane or x-y plane.
The distance between two points formula is further classified into two formulas:
- Distance Between Two Points on a Coordinate Plane
- Distance Between Two Points Formula in 3D Space
Read More: Important MCQs on Coordinate Geometry
Distance Between Two Points on a Coordinate Plane
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The two points lie along the x-axis and y-axis in a two-dimensional plane. The distance of a point from its Y-axis is known as abscissa or X-coordinate and the distance of a point from X axis is known as ordinate or Y-coordinate. The distance between the two points can be calculated by the distance formula.
The formula for calculating the distance (d) for the two given points whose coordinates are (X1, Y1) and (X2, Y2) is:
d = √[(x2 – x1)2 + (y2 – y1)2]
Solved Example
Example: Find the distance between two points P = (1, 2) and Q = (1, 5).
Solution: The distance between two points P = (1, 2) and Q = (1, 5) can be derived by-
d = √[(x2 – x1)2 + (y2 – y1)2]
Substituting the values,
d = √[(1 - 1)2 + (5 – 2)2]
d = 3 units
Read More: Coordinate Geometry Important Questions
Distance Between Two Points Formula in 3D Space
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To find the distance between two points in a 3D space, we need to consider another coordinate that is present in the z-axis. Let us assume to points P (X1, Y1, Z1), and Q (X2, Y2, Z2) in a 3D space.
The distance between for three points is for the 3-D plane-
d = √[(X2 – X1)2 + (Y2 – Y1)2 + (Z2 – Z1)2]
Read More: Coordinates of A Point in Three Dimensions
Solved Example
Example: Find the distance between two points P (4, 5, 8), and Q (4, 9, 8).
Solution: The distance between two points P (4, 5, 8), and Q (4, 9, 8) can be derived by d = √[(X2 – X1)2 + (Y2 – Y1)2 + (Z2 – Z1)2]
d = √[(4 – 4)2 + (9 – 5)2 + (8 – 8)2]
d = √[(0)2 + (4)2 + (0)2]
d = √16
d = 4 units
Derivation of Distance Between Two Points Formula
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To derive the formula to calculate the distance between two points, let us assume there are two points P (X1, Y1) and Q (X2, Y2).
We will assume PQ is a line segment.
Now we will construct a right-angle triangle PQR.
Now applying Pythagoras theorem for the triangle PQR –
PQ2 = PR2 + QR2
d2 = [(x2 – x1)2 + (y2 – y1)2]
d = √[(x2 – x1)2 + (y2 – y1)2]
Thus, the derivation formula is derived.
The Distance Between Two Points Formula for the 3-D plane can be derived similarly.
Read More:
| Related Topics | ||
|---|---|---|
| Polar Coordinates | X and Y Intercepts | Right Angle Triangle Theorem |
| Angle Between Two Planes | Introduction to Three-Dimensional Geometry | Ordinate |
Things to Remember
- Distance between two points refers to the length of the line segment that connects the two given points.
- The distance between two points is always positive.
- Distance between Two Points Formula is used to calculate the distance between two coordinates.
- The distance of a point from its Y-axis is known as abscissa or X-coordinate and the distance of a point from the X-axis is known as ordinate or Y-coordinate.
- Distance Between Two Points Formula for 2D Plane: d =√[(x2 – x1)2 + (y2 – y1)2].
- Distance Between Two Points Formula for 3D Plane: d = √[(X2 – X1)2 + (Y2 – Y1)2 + (Z2 – Z1)2].
Sample Questions
Ques. Find the distance between two points A (2, 10) and B (2, 20). (3 Marks)
Ans. Let us assume
- (X1, Y1) = (2, 10)
- (X2, Y2) = (2, 20)
Now, put the values in the formula –
d = √[(x2 – x1)2 + (y2 – y1)2]
d = √[(2 – 2)2 + (20 – 10)2]
d = √[0 + 100]
d = 10
Ques. A man walks towards the north 20 meters and took a turn towards the east and walked 50 meters more. What is the shortest distance between the two points? (3 Marks)
Ans. The initial point is A and the final point is C. The distance between point A and point B is 20 meters and the distance between point B and point C is 50 meters.
The shortest distance between the initial point A and the final point C is AC.
This distance can be calculated by Pythagoras Theorem-
AC2 = AB2 + BC2
= √(202 + 502)
= √(400 + 2500)
= √(2900) m
Ques. Show the points (2, -1), (0, 1), and (2, 3) are the vertices of a right-angled triangle. (3 Marks)
Ans. Given that,
A = (2, -1)
B = (0, 1)
C = (2, 3)
Let us put the values in the formula-
AB = √[(0 – 2)2 + (1 – (-1))2]
AB = √(4 + 4)
AB = √8
BC =√[(2 - 0)2 + (3 - 1)2]
BC = √(4 + 4)
BC = √8
AC = √[(2 - 2)2 + (3 – (-1))2]
AC =√16
AC = 4
A, B, and C satisfy the Pythagoras theorem
Triangle ABC is a right-angled triangle.
Ques. Find the distance between two points A (1, 5) and B (5, 5). (3 Marks)
Ans. Let us assume that
- (X1, Y1) = (1, 5)
- (X2, Y2) = (5, 5)
Now let us put the values in the formula –
d = √[(x2 – x1)2 + (y2 – y1)2]
d = √[(5 – 1)2 + (5 – 5)2]
d = √[4 + 0]
d = √16
d = 4
Ques. What is the distance between two points? (1 Mark)
Ans. The distance between two points is defined as the length of a line segment that connects two points.
Ques. Find the distance between two points L = (5, 15) and M = (10, 20). (2 Marks)
Ans. The distance between two points L = (5, 15) and M = (10, 20) can be derived by d = root [(x2 – x1)2 + (y2 – y1)2].
d = √[(10 - 5)2 + (20 – 1)2]
d = √50
Ques. How can the distance between two points be calculated in coordinate geometry? (1 Mark)
Ans. Distance between two points can be calculated in coordinate geometry by calculating the length of the line segment joining the two given coordinates.
Ques. What are the coordinates of a point? (2 Marks)
Ans. The points are defined by their coordinates along the X-axis and Y-axis. The coordinates of a point are a pair of values that depicts the exact location of a point. For example, (8, 6) is a coordinate.
Ques. Which formulas are used to calculate the distance between two points? (3 Marks)
Ans. The distance between the two points can be calculated by the distance formula. The formula for calculating the distance (d) for the two given points whose coordinates are (X1, Y1) and (X2, Y2) is:
- For 2-D plane: d =√[(x2 – x1)2 + (y2 – y1)2]
- For 3-D plane: √[(X2 – X1)2 + (Y2 – Y1)2 + (Z2 – Z1)2]
Ques. A boy walks towards the north 40 meters and took a turn towards the west and walked 40 meters more. What is the shortest distance between the two points? (3 Marks)
Ans. The initial point is A and the final point is C. The distance between point A and point B is 40 meters and the distance between point B and point C is 40 meters.
The shortest distance between the initial point A and final point C is AC.
This distance can be calculated by Pythagoras Theorem –
AC2 = AB2 + BC2
= √(402 + 402)
= √(1600 + 1600)
= √3200 m
Ques. Find the distance between two points P (2, 5, 3), and Q (2, 5, 8). (3 Marks)
Ans. The distance between two points P (2, 5, 3), and Q (2, 5, 8) can be derived by d = root [(X2 – X1)2 + (Y2 – Y1)2 + (Z2 – Z1)2]
d = √[(2 – 2)2 + (5 – 5)2 + (8 – 3)2]
d = √[(0)2 + (0)2 + (5)2]
d = √25
d = 5 units
Ques. A girl walks towards the north 30 meters and took a turn towards the east and walked 40 meters more. What is the shortest distance between the two points? (3 Marks)
Ans. The initial point is P and the final point is R. The distance between point P and point Q is 30 meters and the distance between point Q and point R is 40 meters.
The shortest distance between the initial point P and the final point R is PR.
This distance can be calculated by Pythagoras Theorem-
PR2 = PQ2 + QR2
= √(302 + 402)
= √(900 + 1600)
= √(2500)
= 50 m
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