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Now we can talk about the idea of slope, and it really is no exaggeration to say, that the single most important idea to understand about lines in the x-y plane is the idea of slope. This is a tremendously important idea in coordinate geometry. Slope is roughly, a measure of how steep a line is. Different lines are at different angles, and slope is a way of talking about their slants.
One of the most common definitions of slope is rise over run. What does this mean? Suppose we want to find the slope between two points. The run is the horizontal separation from left to right. This is always positive from left to right. The rise is the vertical separation between the two points as we go from left to right.
If the point on the right is higher, then the rise is positive, and if the point on the right is lower, then the rise is negative. And of course if the rise is positive or negative, that means that the slope is positive or negative. Suppose we want to find the slope between the points 2,2 and 6,5. We draw or imagine a little slope triangle.
So this is a small right triangle, the run is the horizontal leg of this triangle, and the rise is the vertical leg. So we see that the rise has a length of three. And the run has a length of 4. So, rise over run is 3/4, 3 quarters. Suppose we want to find the slope between (-4, 2) and (5, -1).
So that would be this. Here’s the slope triangle. As we move from left to right, the line goes down, so the rise is -3. It’s a drop of -3, as we move from left to right. The run is 9. That’s the length of the bottom leg of the triangle.
So the rise over run is –3 over 9, which is –1/3. And notice that –1/3. That suggests that we could also look at it as having a rise of –1 and a run of 3. So in other words over 3, down 1, over 3, down 1, over 3, down 1.
And all of those land on integer points on the line. So that’s another way of thinking about the line. If you are given the numerical coordinates of two points, I would strongly suggest using the slope triangle to visualize the slope. Actually thinking about it visually and finding the slope by finding the rise and the run visually.
Sometimes, you are given an algebraic information and you have to use a formula for slope. But do not make the formula your default. This is one of many example in which over-reliance on the formula produces a shallow mathematical understanding. A much deeper understanding of slope is when you’re thinking about it visually.
Having warned you about the perils of over-reliance on the formula, you do need the formula sometimes so I’ll give you the formula. Suppose we have two general points. So point number one, x1, y1, and point number two, x2, y2. And incidentally, if you’re given two points with numerical coordinates, it doesn’t matter what one you call point one and point two.
You could flip-flip them and you’ll get exactly the same results. So the run is X2 minus X1, the rise is Y2 minus Y1 and the slope is the ratio of these y2- y1 over x2- x1. And again, this is the formula. It’s a handy formula to know, but it should not be the primary way that you think about slope.
You should not be dependent on this formula for finding slope. So, so far, we’ve been talking about the slopes of two individual points. Of course, a line has a slope, and the slope of a line is the slope between any two points on the line. Now, that’s a bit mind boggling if you think about it, because on any line, there’s a continuous infinity of points on that line, but we could pick any two points of that continuous infinity and the slope between them would be the same, and that would be the same as the slope of the line.
If a line has a slope of 1/2, and then we pick any two points on the line, the ratio of rise over run will simplify to 1/2. And really what we have going on here is similar triangles. We have an infinite number of similar triangles. You may remember as we make similar triangles bigger or smaller the ratio stayed the same.
So here are the particular ratio that concerns us is the ratio rise over run. That’s going to be the same no matter how big or small we make the triangles. Let’s think about what a slope of two means M equals 2. So certainly it could mean a rise of 2 and a run of 1, so right 1 and up 2. That’s certainly one thing a slope of 2 means.
But it could mean right k units, and up 2k units. We could go to the right 2, and up 4. To the right 3, up 6. To the right 4, up 8, that sort of thing. Now, also we could do everything backwards. We could go left 1, and down 2 units.
And in a way what we’d be doing there, is we’d have a run of negative 1, and a rise of negative 2. But of course negative 2 divided by negative 1 is still positive 2. So this is still a slope of 2. And extending that, we can go left k units and down 2k units. So it could go left 2, down 4, left 3, down 6, left 4, down 8.
That sort of thing. It’s also important to think about these numerically. Because a slope of 2 means that essentially every time we add one the x coordinate we’re adding two to the y coordinate. And so if we’re starting at -3, -1 then if we add one to the x and add two to the y that brings this to the point -2,1.
Or we could subtract one from the X, and subtract two from the Y, and that would bring us to the point (-4, -3) and so both of those would also be on the line. It’s important to be able to think about slope visually as well as numerically. Now let’s think about a slope of negative two-thirds and everything that, that could mean. Certainly it could mean a run of 3 and a rise of -2, or in other words, right 3 units and down 2 units.
So like this. Or we could multiply that proportionally. Right 3k units and down 2k units. Or we could go right 1 unit and down 2/3 of a unit. And it’s very important to think about this when we’re visualizing say fractions or fractional units.
And we have to do something very precise in the x,y plane. We could also go backwards. We could go left 3 units, and up 2 units. That would also be a slope of negative two-thirds. We could make that proportionately bigger. We could go left 3k units and up 2k units.
Or we could go left 1 unit and up two-thirds of a unit. And all of those are ways to think about a slope of negative two-thirds. Here’s a practice problem. Pause the video and then we’ll talk about this. Okay. A line goes through the point (2, -1) and has a slope of five-thirds.
So find all the points (a, b) on the line where a and b are integers whose absolute values are less than or equal to 10. So, this is essentially a problem in thinking about the numerical way of approaching slope. So, a slope of five-thirds means that we’re gonna add 3 to the x coordinate and add 5 to the y coordinate.
So starting at (2,-1), we’re gonna add 3 to x add 5 to y, and so the first point we get to is (5,4). Do that again we get to (8,9). Do it again, we get to 11,14, but now we’re running into coordinates that have absolute values greater than 10. So that point doesn’t count, but these first three points, those genuinely satisfy the criteria.
Now start at (2, -1) again. And now we’re gonna go the other way. We’re gonna subtract 3 from x and subtract 5 from y. And so that would lead us to (-1, -6). When we do that again, we get to (-4, -11). Well, 11 also has an absolute value greater than 10.
So, we’re done. We found all the points. So in addition to the one point that we were given in the problem, we found these three green points that also satisfy the criteria. Once again, notice it’s very important to be able to think about these numerically without having to visualize it at all.
That’s another very important fact about slope. Notice that if a line has a slope of 1, then the rise equals the run, and the slope triangle is a 45-45-90 triangle. You should know that lines with a slope of 1 or -1 make 45 degree angles with the axes. You don’t have to worry about the exact angles formed by other slanted lines.
Although, it’s good to know that if, say a positive slope is larger than one it makes an angle of more than 45° with the x-axis. Or, if it has a slope of less than one, then it makes an angle of less than 45° with the x-axis. Those are important things to know. Similarly, notice that lines with a slope greater than 1, or less than -1, are steeper than any road that you might walk or drive on.
It’s quite rare to find any road in a town or a city that has a slant of greater than 45 degrees. So a slope of greater than 1 or less than negative 1. These are slopes that you don’t typically see out in the world as far as streets. What is true of the slopes if two lines are parallel? If the lines are parallel, they must rise in sync with the same rise and the same run.
In other words, parallel lines have equal slope. That’s a very important idea. What is true if the slopes of the two lines are perpendicular? First of all, if one goes up, then the other must go down. So in other words, the slopes must have opposite signs. If one is negative, the other must be positive.
Opposite signs is part of the answer, but we need to think about the numerical value of the slope. Here it’s clear for example that the line with the negative slope is much steeper than the line with the positive slope. So they clearly don’t have the same absolute value. Think about what happens when we rotate the slope triangle by 90 degrees.
So our original triangle there is the dark blue triangle. And so we rotate that and we get the light blue triangle. So now what’s tricky. Is that the rise in that light blue triangle, the vertical leg, that’s the original run. The horizontal leg, which would be the run, that was the original rise.
So the rise and the run switch places. What was originally the rise is now the run. What was originally the run is now the rise. That’s what happens when we rotate by 90 degrees. In other words, the numerator and denominator of the slope fractions have switched places.
That’s a reciprocal. So slopes of perpendicular lines are opposite-signed reciprocals, that’s also a very important idea. In other words, if the slope of the original line is slope equals p over q, then the slope of the perpendicular line is negative q over p. So we need to toss in a negative sign and we need to flip the fraction.
If the original slope is positive one half, then perpendicular slope is negative two. So the slope is the rise over run. We find the slope between two points with the slope triangle, or with the slope formula. And I would say use the slope formula only if it’s really algebraic information you’re given, not purely numerical information.
The slope of a line, that’s the slope between any two points on the line. And when you think about that, that’s a bit of a mind-boggling idea, but it’s 100% mathematically true. Lines with slopes of plus or minus 1 make 45 degrees with the axes. Parallel lines have equal slopes and perpendicular lines have slopes that are opposite sign reciprocals of one another.
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