Back to Analytic Geometry
Analytic Geometry:
AG 003 Rectangular coordinates:
By William E. Steinman:
Last time we found we could calculate the distance between two points in our rectangular coordinate system. Another thing we can do is find the point of division of a line in our system.
What the heck does that mean?
The point of division is the point that divides a line segment into a given ratio.
So let’s take our initial coordinate system and slap down a couple of points to define a line segment.
Swell!
Now we can define a third point, call it p(x y) that divides the line so that p1p/pp2 = the ratio r.
We can put that point on our line segment and sketch in some more triangles.
That would look something like what I have below.
We can see that p1m/pn = x – x1/x2 – x = p1p/pp2 = r.
Since we know the ratio, if we know the coordinates of p1 and p2, we can find the coordinates of p. They are given by the equations
x = x1 + rx2/1 – r
y = y1 + ry2/1 + r
That is a lot of diddling to find a point, but it is not complicated. It is all based on things we learned earlier.
Another ting we can do using our triangles and points is find the slope of a line.
Let’s go back to one of our previous sketches.
Here we have an angle of interest at p1.
We define the inclination of our line p1p2 as the smallest positive angle measured in a positive direction from the x axis to our line.
That sounds more complex than it really is.
In our above sketch, that is just the angle between the base of the triangle and the hypotenuse. That is our angle of inclination.
The slope of a line segment is then defied as the tangent of the angle of inclination.
Remember the tangent = side opposite/side adjacent
This means the slope of the line p1p2 is = tan of the angle = (y2 – y1)/(x2 – x1).
That’s enough for this trip.
Go over this stuff to make sure you understand it.
When you get to calculus you will be glad you did.