How to find the distance between two points in an XY plane? GRE Math

How to find the distance between two points in an XY plane?

In coordinate geometry, if you know the coordinates of two points, you can also calculate the distance between them. In fact there is a simple formula for it that you can remember. Let’s say  and  are the two points on the X-Y plane. The distance between A and B can be determined using the distance formula: 

If you find that difficult to remember, you should know that the distance formula is derived in a few simple steps from the Pythagorean Theorem. How? Let’s take a simple example to illustrate this point. Let’s say the points are  and  as shown in the following figure:

Now for a moment consider the segment  to be a hypotenuse of a right triangle that has one side parallel to Y-axis and another side that is parallel to X-Axis:

It’s easy to find the length of the height and the base of this triangle since these sides are parallel to Y and X axis respectively.

Now you can use the familiar Pythagorean theorem to compute the distance between the two points  and .

Three important but amazingly simple things to remember:

1. If the X coordinates of the points are the same, then the distance between them is equal to the difference between the Y coordinates.
2. If the Y coordinates of the points are the same, then the distance between them is equal to the difference between the X coordinates. 
3. If one of the two points is the origin , and the other point has the coordinates , then the distance between the points: 

Example:

Calculate the distance between the points with coordinates  and  .

A) 

B) 

C) 

D) 

E) 

Solution:

Here, 

So, distance, 

 units.

D is the correct answer.