MathematicsChapter 7: Coordinate Geometry
Distance Formula
In the plane, the distance between A(x1, y1) and B(x2, y2) is sqrt((x2-x1)^2+(y2-y1)^2).
Learning objectives
- Explain distance formula in clear mathematical language
- Apply the ideas in from pythagoras
- Solve problems involving order does not matter
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 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.
Mathematical form
y1=y2
x1=x2
Where / symbols
- 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.
Overview
Build a right triangle on the grid, then take the hypotenuse:
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.
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.
Examples A to B same as B to A · (0,0) to (-3,-4) is still 5 · sqrt(2) left as sqrt(2)
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.
Explanation
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.
In simple words
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.
Key points
- sqrt((x2-x1)^2+(y2-y1)^2)
- Pythagoras
- Order does not matter
- horizontal change
- vertical change
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Distance Formula
Sticky Notes · Quick Revision
Core Idea
- In the plane, the distance between A(x1, y1) and B(x2, y2) is sqrt((x2-x1)^2+(y2-y1)^2).
Key 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.
Main Pattern
- 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.
Worked Reminder
- (1,2) to (4,6) is 5
- (0,0) to (3,4) is 5
- (2,1) to (2,1) is 0
Common 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 matter, but mixing x of one point with y of the other does matter.
Exam Tip
- State the rule for distance formula, show each algebraic step, and check the final answer against the question.
60-second revision
- 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|.
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