Конспект: формула відстані в Декартових координатах

Usually, you can't make difficult things in life disappear and replace them with easy things, but in the case of finding distances, you can! Let's just think about right triangles for a second. We all like right triangles, because we know a lot about them. Now let's go back to thinking of distances on a coordinate plane.

So we have this line segment and we want to find its length. Again, let's go back to thinking about right triangles and see if we can find one in this diagram.

See it now?? Now all we have to do is Pythagorean Theorem to find the length of the line segment (which is now the hypotenuse!)

And now you have your answer!!

This process has been condensed into a formula known simply as the Distance Formula.

This may look hard, but always remember that it is just Pythagorean Theorem!!!

For example, in our question, we could have used this formula.

Distance between two points

Finding the distance

Imagine that we are given two points on the Cartesian plane, $A=(2,1)$ and $B=(5,3)$. We need to determine the length of the blue line segment $AB$ joining these two points.

In real life, we could just use a ruler, but that wouldn't give us an exact measurement.

So, what do we do?

We can measure horizontal or vertical distances easily, since we can just look at the difference between $x$ or $y$ coordinates.

In this example, the horizontal distance (which I will call $d_H$) between $A$ and $B$ is just the difference in their $x$ values: $$d_H = 5-2 = 3$$

The vertical distance ($d_V$) is the difference in their $y$ values:

These lengths are represented by the dotted green line segments in the diagram below:

We don't know what the distance between $A$ and $B$ is yet, but we now have a right-angled triangle with two sides.

By the Theorem of Pythagoras, we know how to calculate the length of the hypotenuse... and the hypotenuse is the same as $AB$. This means that we can find the distance:

Let $d=$ the distance between $A$ and $B$. Then:

Finding a formula

If we had to use the method above every time we wanted to find the distance between two points, we'd actually be wasting a bit of time. We can work out a formula that we can use instead.

To do this, we generalise. Let $A=(x_1,y_1)$ and let $B=(x_2,y_2)$.

Then we know that the horizontal distance between $A$ and $B$ is $x_2-x_1$ and the vertical distance between them is $y_2 - y_1$. (If you struggle to see this, compare the general diagram below to diagram with numbers further up in the explanation.)

From this, we continue using the same method as before to find $d_{AB}$, the distance between $A$ and $B$:

This last line gives us an expression to find the distance between two points $A$ and $B$ when we know their coordinates.

Use the formula to find the distance between pairs of coordinates given in each case:

Considering a Cartesian system where we need to find the distance between two points namely $A(x_A,y_A)$ and $B(x_B,y_B)$.

We can now draw the following lines parallel to x axis and y axis from these points to obtain a right triangle , where hypotenuse is the distance between the two points as shown:

Thus, substituting the coordinates to the above figure. We can find the length of the sides of the triangle as follows.

Now, using the famous PYTHAGORAS THEOREM where: $$(Hypotenuse)^2 = (Height)^2 + (Base)^2$$ $$(Hypotenuse)^2 = (y_B-y_A)^2 + (x_B-x_A)^2$$

Thus,

Distance between two points = $\sqrt{(y_B-y_A)^2 + (x_B-x_A)^2}$

Cheers!

We have two points here, $A(x_1,y_1)$ and $B(x_2,y_2)$, and we need to find the distance between them.

How can we do this? Well, let's start with a sample problem.

Okay, we have the points $(x_1,y_1)$ and $(x_2,y_2)$ We would know how to do this if one of the points was $(0,0)$. Just do $\sqrt{(height)^2+(length)^2}$. Say, why don't we MAKE one of the points $(0,0)$? Translating the first point $(x_1,y_1)$ to $(0,0)$, we also have to subtract $x_1$ from $x_2$, and $y_1$ from $y_2$ (this will not affect the distance). So, our new points are $(x_2-x_1,y_2-y_1)$, and $(0,0)$. Therefore, the length (leg 1) is $x_2-x_1$, and the height (leg 2) is $y_2-y_1$. So, the distance between $(x_1,y_1)$ and $(x_2,y_2)$ is $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$ (also notice that it's the same thing $\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$).