Coordinate Geometry of a Straight Line

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Welcome to class! 

In today’s class, we will be talking about coordinate geometry of a straight line. Enjoy the class!

Coordinate Geometry of a Straight Line

Any straight line has an equation of the form

y = mx + c

where m, the gradient, is the height through which the line rises in one-unit step in the horizontal direction

c, the intercept, is the y-coordinate of the point of intersection between the line and the y-axis. This is shown in the figure below.

The straight line, y = mx + c

If we know the gradient m of a straight line with unknown intercept c, and the coordinates (x₁ , y₁) of a point through which it passes, then we know that

y₁ = mx₁ + c

and therefore

c = y₁ − mx₁

If we substitute into

y = mx + c

we obtain

y = mx− mx₁ + y₁

which we can rearrange to give

y− y₁ = m(x − x₁)

If we know two points (x₁ , y₁) and (x₂ , y₂) through which passes a line with unknown gradient m and intercept c, then

y₁ = mx₁ + c ,

y₂  = mx₂ + c

Subtracting the first equation from the second gives

y₂ − y₁ = m(x₂ − x₁)

and therefore

$m =(x2 – x1)/(y2 – y1)$

The equation of the line is therefore

$y - y1 = (y2 – y1)/(x2 – x1 ) (x - x1)$

The midpoint of the line joining two points

Once we know the coordinates of two points on a straight line, we can find the mid-point of the line.

Distance between two points

The distance formula is derived from the Pythagoras theorem. To find the distance between two points, if A is (x₁ , y₁) and B is (x₂ , y₂), all that you need to do is use the coordinates of these ordered pairs.

The distance between two points is given by:

$Distance=\surd (〖(x2–x1){〗}^2+〖(y2–y1\left){〗}^2\right)$

 

In our next class, we will be talking more about the Coordinate Geometry of straight line.  We hope you enjoyed the class.

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