Distance Between Two Points On A Graph

The distance between two points on a Cartesian plane is the length of the straight line that joins the two points. Hence, you can use strategies to get the length of the straight line to solve these types of problems.





If Two Points Have Common Y Coordinates



If you have 2 points with the same y coordinates, these 2 points will form a horizontal straight line. For example:



To obtain the distance between two points, subtract the x coordinate of one point off the other, and discard the "-" sign (that is, get the absolute value), if present.


Example 1
From the graph above, find the distance between points:
(a) A and B
(b) C and D
(c) E and F

Solution
(a) Distance of AB = |-4 – 3| = |-7| = 7 units
(b) Distance of CD = |-4 – (-2)| = |-4 + 2| = |-2| = 2 units
(c) Distance of EF = |1 – 4| = |-3| = 3 units

If you're given a graph, the easier alternative is to just count the number of squares sides that the line touches. Multiply that number by the number of units each side represents (in this case, the unit per square is 1).


Example 2
Calculate the distance between each pair of points given below.
(a) A(4, 3) and B(6, 3)
(b) C(1, -5) and D(-5, -5)

Solution
(a) Distance of AB = |4 – 6| = 2 units
(b) Distance of CD = |1 – (-5)| = |1 + 5| = 6 units


If Two Points Have Common X Coordinates

If you have 2 points with the same x coordinates, these 2 points will form a vertical straight line. For example:



To obtain the distance between 2 points, subtract the y coordinate of one point off the other, and discard the "-" sign (that is, get the absolute value), if present.


Example 1
From the graph above, find the distance between points:
(a) A and B
(b) C and D
(c) E and F

Solution
(a) Distance of AB = |-1 – (-4)| = |-1 + 4| = 3 units
(b) Distance of CD = |3 – (-2)| = |3 + 2| = 5 units
(c) Distance of EF = |4 – 2| = 2 units


Example 2
Calculate the distance between each pair of points given below.
(a) P(-1, -1) and Q(-1, 5)
(b) R(6, -6) and S(6, -2)

Solution
(a) Distance of PQ = |-1 – 5| = 6 units
(b) Distance of RS = |-6 – (-2)| = |-6 + 2| = |-4| = 4 units


Using Pythagoras' Theorem



Pythagoras' theorem states that the square of the hypotenuse (the longest side, c) of a right angled triangle is the sum of the squares of the other 2 sides.

i.e.



So the theorem states that c² = a² + b². You're going to use this knowledge to find the distance between two points on a Cartesian plane.

Say you're given 2 coordinates with different x coordinates and y coordinates respectively. The 2 points can be joined by a line, which can then be imagined to form a right triangle.




Example 1
Based on the graph above, use Pythagoras theorem to find the distance between point A and B.

Solution
Distance of YZ = |2 – 5| = 3 units
Distance of XZ = |3 – (-2)| = 5 units

Using Pythagoras' theorem,


Using a Formula

Of course, there's an algebraic formula to represent the distance between 2 points:

For any 2 points represented by:

In essence, this is Pythagoras' theorem expressed as an algebraic equation. If you use it, you'll find that the steps are identical to using the theorem directly.


Example 1
Find the distance between the points A(2, -2) and B(-4, -5).

Solution



Example 2
Find the possible values of p if the distance between two points A(1, 2) and B(p, 14) is 13.





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