Last week, in Fundamental Understanding of Mathematics LVI, we took a look at drawing a line on a two dimensional Cartesian coordinate system (aka, a graph) and saw how the slope of the line came from the rise and run of that line. We also found that the slope could be a rate, depending on the units we used for the axes of the graph. Today, we are going to start exploring some of a graph's geometric connections.
Even though we've been talking about numbers, when we put those numbers on a graph, they form a drawing. So far, the drawing has been pretty simple: a line. But we aren't limited to drawing just lines.
The breakthrough Rene Descartes made when he designed this coordinate system is that it connects algebra (using numbers and calculations) and geometry (using drawing instruments and logic.) In order to make it more interesting, we will use a more complex geometric shape: a triangle.
The triangle can be defined by the three points at its corners, or vertices. Since we drew the corners (the red dots) on a plane with Cartesian coordinates, we can also use the coordinates of the corners to define the triangle:
Triangle (4,7),(5,2),(12,6)
On the one hand, the triangle is a shape drawn on the graph using three line segments, on the other hand, the triangle is a set of three ordered pairs of numbers, which say where the corners of the triangle would be, when those three ordered pairs are placed on the graph (remember, by convention, the x number is given first, the y number is second.)
When I was in elementary school, and probably when you were, too, our teacher got us to cut those triangle out, and we slid them around on our desk, and put them on top of other triangles to see whether they were the same. Later on, in Middle or High School, we were taught that sliding triangles around was called "translation" and if two triangles fit on each other they were "congruent."
Testing a triangle for congruence might involve more, though, than just sliding it around. Sometimes we rotated the triangle, or flipped it over. We might save those complicated moves for later, but today, let's take a look at sliding the triangle around.
One of the nice things about moving triangles in a computer drawing program is that they don't accidentally rotate, which might happen if you move a cut out triangle by hand. (One of the not so nice things is that any minor imperfections in drawing the graph show up when you do that. Please pretend that rightmost point is where the lines cross and not slightly below.)
This triangle now has new corner points. It has become
Triangle (7,10),(8,5),(15,9)
Let's compare the two triangles
Blue Triangle (7,10),(8,5),(15,9)
Red Triangle (4, 7),(5,2),(12,6)
There is a pattern here, and you might see it looking at the two triangles, or you might see it by examining the pairs of numbers. On the graph the Blue Triangle is higher (moved in the positive Y direction) and further to the right (moved in the positive x direction.) In the ordered pairs of numbers, the Blue Triangle numbers are higher that the corresponding Red Triangle numbers.
If we take a closer look at how we moved each corner, we notice that the rise and the run from each Red Triangle corner to the corresponding Blue Triangle corner is the same, rise is 3 and run is 3.
If we take Red Triangle's ordered pairs (let's do the highest corner) (4,7) where 4 is along the x axis, and 7 is along the y axis, and add the run to the x axis number, and the rise to the y axis number, we get
(4+3 , 7+3) = (7,10)
which is the highest corner of the Blue Triangle.
The same is true for the other two corners. If we add rise and run to the ordered pair for each corner of the Red Triangle, we'll get the corners of the Blue Triangle.
So, to move a drawing around using arithmetic, we choose the direction and distance we want to move the drawing. Then we find the rise and run for that move (if we draw an arrow showing the direction and distance, the rise and run can be found by subtracting the feather end of the arrow from the pointy end) and add the rise to all the y numbers in all the points in our drawing's ordered pairs, and add the run to all the x numbers in all the points in our drawing, as well.
Have fun in the comments.