Distance between two points

Distance between two points
The distance between two points A(a, b), B(c,d) is equal to
…(1)
The distance of a point P(x,y) from the origin is equal to
…(2)
The horizontal leg of a right triangle is |c−a|.
The vertical leg of the right triangle is |d−b|.
Using the Pythagorean Theorem (*) to find the length of the hypotenuse,
AB2=|c−a|2+|d−a|2=(c−a)2+(d−a)2
AB= →(1)
(As the squares become always positive value (or zero), it does not matter whether the signs of c−a and d−b are positive or negative.)

The distance of a point P(x,y) from the origin :
In (1), assign x,y,0,0 to a,b,c,d
OP== →(2)
Example (1)
The distance between two points A(1, 1), B(4,5) :
The difference in x-coordinates is 4−1=3.
The difference in y-coordinates is 5−1=4.
Using the Distance Formula (1),
AB===5
The distance between two points A(−3, 4), B(2, −1) :
The difference in x-coordinates is 2−(−3)=5.
The difference in y-coordinates is −1−4=−5.
(It does not matter whether the sign is positive or negative.)
Using the Distance Formula (1),
AB===5
Example (2)
The distance of a point P(3, 2) from the origin :
Using the Distance Formula (2),
AB==
(*) Pythagorean Theorem
In a right triangle, let c be the length of the hypotenuse, a and b be the lengths of the two legs.
Then
a2+b2=c2










Notice:
When a=c or b=d, the line becomes only a vertical or horizontal line segment but the formula is true.
The distance between two points A(2, 1), B(2, 4) is only the difference between the y-coordinates :
AB=4−1=3
From the Formula
AB===3
These are equal.











Common mistakes
• Subtraction should not be done between the coordinates of one point, but should be done between a pair of x-coordinates and a pair of y-coordinates.
For example, in order to find the distance between two points A(1, 2), B(3, 4)
← Incorrect
← Correct
• Common mistakes are made while subtracting a negative number.
For example, in order to find the distance between two points C(−5, 6), D(7, 8)
← Incorrect
← Correct
QuestionFind the distance between the following two points.
(Choose from the right column.)
['+str_counter0+' / '+q_set0.length+']BackNext
Distance

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