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Distance Formula
In the plane, the distance between A(x1, y1) and B(x2, y2) is sqrt((x2-x1)^2+(y2-y1)^2). The formula is Pythagoras: the run x2-x1 and the rise y2-y1 are the legs of a right triangle, and AB is the hypotenuse. Order does not matter because each difference is squared. Distance is never negative: the square root of a sum of squares is at least 0.
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Distance Formula
What is it?
In the plane, the distance between A(x1, y1) and B(x2, y2) is sqrt((x2-x1)^2+(y2-y1)^2). The formula is Pythagoras: the run x2-x1 and the rise y2-y1 are the legs of a right triangle, and AB is the hypotenuse. Order does not matter because each difference is squared. Distance is never negative: the square root of a sum of squares is at least 0.
In simple words
- In the plane, the distance between A(x1, y1) and B(x2, y2) is sqrt((x2-x1)^2+(y2-y1)^2).
- From A to B, the horizontal change is x2-x1 and the vertical change is y2-y1.
- Order does not matter: (x1-x2)^2 equals (x2-x1)^2, and the same for y.
- The number-line case is hidden inside: if y1=y2, the formula collapses to |x2-x1|.
Important terms
sqrt((x2-x1)^2+(y2-y1)^2): In the plane, the distance between A(x1, y1) and B(x2, y2) is sqrt((x2-x1)^2+(y2-y1)^2).
Pythagoras: The formula is Pythagoras: the run x2-x1 and the rise y2-y1 are the legs of a right triangle, and AB is the hypotenuse.
Order does not matter: Order does not matter because each difference is squared.
horizontal change: From A to B, the horizontal change is x2-x1 and the vertical change is y2-y1.
What you should learn
- Explain distance formula in clear mathematical language
- Apply the ideas in from pythagoras
- Solve problems involving order does not matter
Examples
- (1,2) to (4,6) is 5
- (0,0) to (3,4) is 5
- (2,1) to (2,1) is 0
FROM PYTHAGORAS
From A to B, the horizontal change is x2-x1 and the vertical change is y2-y1. Those two changes meet at a right angle, so Pythagoras says AB^2 = (x2-x1)^2 + (y2-y1)^2. Take the principal square root to get AB = sqrt((x2-x1)^2+(y2-y1)^2). If the points are (1,2) and (4,6), run=3, rise=4, distance=5.
ORDER DOES NOT MATTER
Order does not matter: (x1-x2)^2 equals (x2-x1)^2, and the same for y. You may label either point as the first. That is the plane version of |b-a| = |a-b|. If you expand the squares, a minus sign disappears. Leave the answer under a sqrt unless it is a perfect square.
Recommended for Class 9
Full school-level explanation
In the plane, the distance between A(x1, y1) and B(x2, y2) is sqrt((x2-x1)^2+(y2-y1)^2). The formula is Pythagoras: the run x2-x1 and the rise y2-y1 are the legs of a right triangle, and AB is the hypotenuse. Order does not matter because each difference is squared. Distance is never negative: the square root of a sum of squares is at least 0.
Build a right triangle on the grid, then take the hypotenuse:
What it means
In the plane, the distance between A(x1, y1) and B(x2, y2) is sqrt((x2-x1)^2+(y2-y1)^2).
Main ideas
1. FROM PYTHAGORAS: From A to B, the horizontal change is x2-x1 and the vertical change is y2-y1. 2. ORDER DOES NOT MATTER: Order does not matter: (x1-x2)^2 equals (x2-x1)^2, and the same for y.
Why it matters
The number-line case is hidden inside: if y1=y2, the formula collapses to |x2-x1|. If x1=x2, it collapses to |y2-y1|. Both are still Pythagoras with one leg zero.
How to answer in an exam
Write sqrt((x2-x1)^2+(y2-y1)^2) with both squares inside. A frequent error is to square after adding, or to forget one of the two differences. Order does not matter, but mixing x of one point with y of the other does matter.
FROM PYTHAGORAS
From A to B, the horizontal change is x2-x1 and the vertical change is y2-y1. Those two changes meet at a right angle, so Pythagoras says AB^2 = (x2-x1)^2 + (y2-y1)^2. Take the principal square root to get AB = sqrt((x2-x1)^2+(y2-y1)^2). If the points are (1,2) and (4,6), run=3, rise=4, distance=5.
Remember
- horizontal change
- vertical change
- sqrt((x2-x1)^2+(y2-y1)^2)
Examples
- (1,2) to (4,6) is 5
- (0,0) to (3,4) is 5
- (2,1) to (2,1) is 0
ORDER DOES NOT MATTER
Order does not matter: (x1-x2)^2 equals (x2-x1)^2, and the same for y. You may label either point as the first. That is the plane version of |b-a| = |a-b|. If you expand the squares, a minus sign disappears. Leave the answer under a sqrt unless it is a perfect square.
Remember
- Order does not matter
- (x1-x2)^2
- perfect square
Examples
- A to B same as B to A
- (0,0) to (-3,-4) is still 5
- sqrt(2) left as sqrt(2)
Remember
- sqrt((x2-x1)^2+(y2-y1)^2)
- Pythagoras
- Order does not matter
- y1=y2
- |x2-x1|
- both squares inside
- Order does not matter
- horizontal change
How the ideas connect
The number-line case is hidden inside: if y1=y2, the formula collapses to |x2-x1|. If x1=x2, it collapses to |y2-y1|. Both are still Pythagoras with one leg zero.
Deeper understanding
Go beyond memorising Distance Formula. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
The number-line case is hidden inside: if y1=y2, the formula collapses to |x2-x1|. If x1=x2, it collapses to |y2-y1|. Both are still Pythagoras with one leg zero.
Common mistakes
Exam focus
Core Idea
Key Terms
Main Pattern
Worked Reminder
Exam focus
Quick revision
- sqrt((x2-x1)^2+(y2-y1)^2)
- Pythagoras
- Order does not matter
- y1=y2
- sqrt((x2-x1)^2+(y2-y1)^2): In the plane, the distance between A(x1, y1) and B(x2, y2) is sqrt((x2-x1)^2+(y2-y1)^2).
- Pythagoras: The formula is Pythagoras: the run x2-x1 and the rise y2-y1 are the legs of a right triangle, and AB
Write sqrt((x2-x1)^2+(y2-y1)^2) with both squares inside. A frequent error is to square after adding, or to forget one of the two differences. Order does not matter, but mixing x of one point with y of the other does matter.
For long answers on Distance Formula: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- sqrt((x2-x1)^2+(y2-y1)^2)
- Pythagoras
- Order does not matter
- y1=y2
Important formulas
- sqrt((x2-x1)^2+(y2-y1)^2)
- Pythagoras
- Order does not matter
Common exam traps
- Write sqrt((x2-x1)^2+(y2-y1)^2) with both squares inside.
- A frequent error is to square after adding, or to forget one of the two differences.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Write sqrt((x2-x1)^2+(y2-y1)^2) with both squares inside.
- Mistake
A frequent error is to square after adding, or to forget one of the two differences.
- Mistake
Do not mix up from pythagoras with order does not matter; use the condition given in the question.
- Mistake
Write sqrt((x2-x1)^2+(y2-y1)^2) with both squares inside. A frequent error is to square after adding, or to forget one of the two differences. Order does not ma
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
sqrt((x2-x1)^2+(y2-y1)^2)
Pythagoras
Order does not matter
y1=y2
In the plane, the distance between A(x1, y1) and B(x2, y2) is sqrt((x2-x1)^2+(y2-y1)^2). The formula is Pythagoras: the run x2-x1 and the rise y2-y1 are the leg
y1=y2
- Definition
- In the plane, the distance between A(x1, y1) and B(x2, y2) is sqrt((x2-x1)^2+(y2-y1)^2). The formula is Pythagoras: the run x2-x1 and the rise y2-y1 are the leg
- Formula
- y1=y2
- Example
- (1,2) to (4,6) is 5
- Critical warning
- Write sqrt((x2-x1)^2+(y2-y1)^2) with both squares inside.
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Exam Practice
Define
Q1. Define Distance Formula.
Explain
Q2. Explain: FROM PYTHAGORAS.
Example
Q3. Give one clear example of Distance Formula and say why it fits.
Mistake
Q4. State one common mistake students make about Distance Formula.
Exam note
Q5. Write a short exam note on Distance Formula: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.