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Coordinate plane – The midpoint and distance formula

Coordinate plane is a plane such that each point is uniquely specified by a pair of numerical coordinates. In the plane there are two vertical lines, called axis, the horizontal one is the x-axis and the vertical one is the y-axis. Their intersection is called the point of origin.

Coordinate is an ordered pair which tells us the location of the point it represents. The first number in the ordered pair tells us how far the point is located from the point of origin on the x-axis, and the other number how far on the y-axis.

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

The point of origin is always (0, 0).

The two axes divide the plane into four sections called quadrants. The quadrants are labelled with Roman numerals, starting at the positive x-axis and going around anti-clockwise. All the points in the first quadrant have both coordinates positive, in the second quadrant all the x coordinates are negative and y coordinates positive, in the third both coordinates are negative, and in the fourth the x coordinates are positive and y coordinates negative.

For example, A(1,1) is a point in the first quadrnat, B(-3,2) in the second quadrant, C(-4,-4) in the third quadrant and D(5,-2) in the fourth.

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

x coordinate is found by drawing a line through A perpendicular to x-axis, and y coordinate is found by drawing a line through A perpendicular to y-axi. The vertical line intersects the x-axis in 2, and horizontal line y-axis in 3 which means that the x-coordinate of A is 2, and y-coordinate of A is 3. This is writen as A(2, 3).

The midpoint formula

How does one calculate the exact middle of the segment between A and B, if for example A(2, 4) and B (-4, 2).

Midpoint of a segment is halfway between the two end points of the segment. Midpoint will also have a x and y coordinate. Its x coordinate is halfway between the x coordinates of the end points, and its y coordinate is halfway between the y coordinates of the end points. To calculate it add modules of x coordinates and divide by 2, and the same for y coordinates.

In the previous example: \frac{-4 + 2}{2} = -1, \frac{4+2}{2} = 3.

The exact formula is:

\ M = (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})

where x_1 and y_1 are coordinates of the first point, and x_2 and y_2 of the second.

The distance formula

The distance between points A and B is marked with a modul: |AB|. And represents the shortest lenght from one point to another.

For example, if A(-2, -2), B( 4, -2) and C(4, 2), then the distance between A and C is easy to determine since 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: |ACw|= 2 + 4 = 6. Similarly C and B share the x coordinate so: |CB|= 2 + 2 = 4.

 

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

So |AB|^2 = |AC|^2 + |CB|^2

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

|AB|^2 = \sqrt{52}

However this is a longer way, so let discuss a shorter one.

If two points \ P_1(x_1, y_1) and \ P_2(x_2, y_1) are two points:

\mid P_1 P_2 \mid^2 = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

It doesn’t matter which point is used as P_1 and which as P_2.

For example, \ A(-2,2) and 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, which is a number that describes steepness of the line, or how much it angles from x-axis. It is usually denoted by “m“.

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

A slope is calculated by the formula:

m = \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 1:

If \ 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}

The slope of this line equals to -\frac{3}{14}.

Example 2:

Calculate the slope of the 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}

Example 3:

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 from that follows that the lines are parallel.

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Slope

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