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100 Day Challenge 2020

Parallel or Not?

How can we tell if two lines are parallel? Lines go on forever, so it isn’t possible to examine an entire line to be sure that it doesn’t intersect the other. But there are more practical ways to tell, and the best method depends on how the lines are presented.

When lines are given as equations, we can compare their slopes. When lines are given visually, it’s not always as straightforward to determine if they’re parallel. But it can be easier to determine that they’re not parallel. For a look at how, keep reading; otherwise, try to find a pair of non-parallel lines in the challenge below.

Let’s look at a pair of parallel lines, y=12x+1{y=\frac 1 2 x + 1} and y=12x+3.{y=\frac 1 2 x + 3.} Both of these lines are given in slope-intercept form y=mx+b,y=mx+b, where mm is the slope and bb is the yy-intercept. Since the slope is m=12m=\frac 1 2 for both and they have different yy-intercepts, we know that they’re parallel.

On the graph above, we have two vertical distances between the lines highlighted. On the left, the distance is between the points on each line with the xx-coordinate x=2.x=2. On the right, it’s between the points with x=4.x=4. In both cases, the vertical distance is 2,2, the difference between the yy-coordinates. In fact, between any two points with the same xx-coordinate (one on each line), the vertical distance will be 2.2.

Why? We can generalize this for any pair of parallel lines. Consider y=mx+b1{y=mx+b_1} and y=mx+b2,{y=mx+b_2,} with b1b2.b_1 \neq b_2. If we look at a pair of points, one on each line, with the same xx-coordinate, they will necessarily have different yy-coordinates — let’s call them y1y_1 and y2,y_2, respectively. Then the vertical distance between the points is y1y2.|y_1-y_2|. We can calculate what that distance will be: y1y2=(mx+b1)(mx+b2)=b1b2. \begin{aligned} |y_1-y_2| &= \big|(mx+b_1) - (mx+b_2)\big| \\ &= |b_1 - b_2|. \end{aligned} The mxmx terms cancel since the xx-coordinates are the same. We see that whenever we have two parallel lines, the vertical distance between points with the same xx-coordinate is always the same.

Is the same true for horizontal distances between points with the same yy-coordinate? On the graph below, we have two horizontal distances highlighted. The lower one is between the points on each line with the yy-coordinate y=3,y=3, and the higher one is between the points with y=4.y=4.

In both cases, the horizontal distance is 4,4, the difference between the xx-coordinates. Similarly to the vertical case, between any two points with the same yy-coordinate (one on each line), the horizontal distance will be 4.4.

Again, we can prove this by generalizing with the pair of parallel lines y=mx+b1{y=mx+b_1} and y=mx+b2,{y=mx+b_2,} with b1b2.b_1 \neq b_2. If we look at a pair of points, one on each line, with the same yy-coordinate, they will necessarily have different xx-coordinates — let’s call them x1x_1 and x2,x_2, respectively. Then the horizontal distance between the points is x1x2.|x_1-x_2|. To solve for that, note that since the yy-coordinates are the same, we can set them equal: mx1+b1=mx2+b2. mx_1+b_1 = mx_2+b_2. Subtracting mx2mx_2 and b1b_1 from each side, we get mx1mx2=b2b1, mx_1 - mx_2 = b_2 - b_1, and factoring out the mm on the left side gives m(x1x2)=b2b1. m(x_1-x_2) = b_2 - b_1. Now, assuming m0m \neq 0 (why is that a special case?), we divide by mm on each side, which makes x1x2=b2b1m. |x_1-x_2| = \left| \frac{b_2-b_1}{m} \right|.

So, whenever we have two parallel lines ((with slope m0),m \neq 0), the horizontal distance between points with the same xx-coordinate is always the same.

To summarize, when lines are parallel, the vertical distance between points with the same xx-coordinate and the horizontal distance between points with the same yy-coordinate are both constants. Can you use this property to determine when lines aren’t parallel? That’s today’s challenge.

Today's Challenge

Which pair(s) of lines are parallel?

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