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Coordinate plane – The midpoint and distance formula

Coordinate plane is a two dimensional number line. It is composed of two vertical line-axis: the horizontal one, x-axis, and vertical, y-axis. Their intersection is called the point of origin.

A coordinate is an information that tells us locations of the points we are observing. It is made out of two numbers, first number tells us how far is the point located from point of origin on the x-axis, and the other one, from y-axis.

Both axis have a positive and negative side. Negative side of x-axis is found on the left from point of origin, and positive on the right. Negative side of y-axis is found down from the point of origin, and positive up.

Point of origin is always in (0, 0).

How to determine coordinates of specific points located in a coordinate plane?

Let’s see it on an example where you have to find coordinates for point A. Numbers on the number line indicate how large your basic measure is. Every segment between two points that show you how far you are from origin point is of constant lenght.

Now, back to finding coordinates. Coordinates always come in form of ordered pair. So A will have coordinates A(x, y).

X coordinate is found by drawing a perpendicular line from A to x-axis. And y coordinate is found by drawing a horizontal line from A to y-axis. Their intersection with axes will show you your coordinates.

point A in a coordinate plane

The vertical line intersects x-axis in 2, and horizontal line y-axis in 3. That means that x-coordinate of A will be 2, and y-coordinate of A will be 3.

We write this as: \ A(2, 3).

number on the line

The midpoint formula

Let’s say you drew two points in the coordinate plane, and conected them with a straight line. You got a line segment and wanted to know how to accurately find the exact middle of your segment. Your points were: A(2, 4) and B (-4, 2).

two point in coordinate plane
First, how do you calculate total distance your line crossed on individual axes? You simply count it, it does not matter if you are on the negative or positive side, their distance is always positive.

Or you can add distance from origin to the point one one side, and add distance from the origin to the point on the other side. So, the total distance your line crossed on x-axis is \ 4+ 2 = 6.

And if both points are on the positive or negative side, you have to subtract the lowest coordinate from the highest. The total distance your line crossed on y-axis is \ 4 - 2 = 2.

You can always find those numbers simply by counting how many basic measures exist between x or y coordinates of your points.

Midpoint of a segment is a point that is located exactly in the middle of your segment. Midpoint will also have a x and y coordinate. X coordinate will be the number you get when you add x coordinates from your points and divide them by two. In this case that number will be \frac{-4 + 2}{2} = -1.

And y coordinate will be the number you get when you add y coordinates from your points and divide them by two. In this case that number will be \frac{4+2}{2} = 3

graphical solution of midpoint formula

The exact formula is: \ M = (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}) where x1 and y1 are coordinates from your first point, and x2 and y2 from second.

The distance formula

Now you drew three points in a coordinate plane. And you connected them with a straight lines, and drew a triangle. Now you are wondering how to find out the exact distance between each two points. The distance between points A and B is marked with a modul: \mid AB \mid. And represents the shortest lenght from one point to another.

Let’s say you drew points \ A(-2, -2), B( 4, -2) and \ C(4, 2).

distance formula

The distance between A and C is easy, because their y- coordinates are the same so all you have to do is add length from the left to the one on the right. That means that \mid AC \mid = 2 + 4 = 6.

Also the distance between C and B is similar, only now they share common x-coordinate. That means that \mid CB \mid = 2 + 2 = 4.

But what to do with distance between A and B? As you maybe already noticed, triangle ABC is a right angled triangle, and the distance between A and B is the length of its hypotenuse. Now that we know that we can easily calculate our distance from pythagorean theorem.

So \mid AB \mid^2 = \mid AC \mid^2 + \mid CB \mid^2

\mid AB \mid^2 = 36 + 16 = 52 / \sqrt

\mid AB \mid^2 = \sqrt{52}

But this way is too slow, you would always first have to find your triangle, calculate distance from all other points, and then use pythagorean theorem.

If you have two points \ P1(x1, y1) and \ P2(x2, y1):

\mid P1 P2 \mid^2 = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

And it does not matter which point you use as P1, and which for P2.
Let’s try it out on our points. \ A(-2,2), B(4,-2)

\mid P1 P2 \mid^2 = \sqrt{(4 - 2)^2 + (-2 - (-2))^2} = \sqrt{2^2 + (-2 + 2)^2} = \sqrt{4 + (0)^2} = \sqrt{4} = 2

Parallel lines in the coordinate plane

Every line has a slope. The slope of a line is a number that describes steepness of the line, or how much it angles from x-axis.

If the slopes of two lines are equal, then the lines are parallel.

A slope is calculated by the formula:

Slope = \frac{y_2 - y_1}{x_2 - x_1}

Where \ x_1, y_1 and \ x_2, y_2 are any two points on that line.

Example: Let’s say we know two points on our line: \ A(-4, 5) and B(\frac{2}{3}, 4). Calculate the slope.

Slope is usually marked with an “m”.

m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 5}{\frac{2}{3} - (-4)} = \frac{-1}{\frac{14}{2}} = - \frac{3}{14}

So, the slope of this line equals to -\frac{3}{14}.

Example 2: Calculate the slope of the given line.

calculating slope of given line

First you choose two points trough which the line goes trough. For instance, we’ll use (-2, 0) and (0, 1).

m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 0}{0 - (-2)} = \frac{1}{2}

As we already said, if two lines are parallel, their slopes are the same.

Example:

If the first line contains points A(2, 4) and B(3, 2), and second line contains points C(4,8) and D(6, 4), are those lines parallel?

m_1 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 4}{3 - 2} = \frac{2 - 4}{3 - 2} = -2

m_2 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 8}{6 - 4} = \frac{-4}{2} = -2

Their slopes are equal, which indicates that the lines are parallel.

parallel lines

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