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Similar triangles & slope: proportion of segments | Grade 8 (TX) | Khan Academy

2m read
·Nov 10, 2024

We're told triangle PQR and triangle ABC are similar triangles. Which proportion shows that the slope of PR, right over here, equals the slope of AC?

So pause this video and see if you can figure that on your own before we do this together.

All right, well, let's just think about how we can express each of these slopes in terms of the other sides of this triangle. We know that slope is change in y, so slope is change in y over change in x. If we were to say to go from point A, right over here, to point C, what is our change in y? Well, our change in y is the length of this line right over here. So our change in y is going to be the length of segment AB, which we could just write as AB.

And then our change in x is going to be the length of segment BC. That's our change in x. So it's AB over BC. Now, let's do the same thing for this triangle right over here. I'll do it a different color. What's our change in y? Well, our change in y is the length of segment PQ.

So let me write this: change in y over change in x. Our change in y is the length of segment PQ. When we don't write the little line segment on top, this means we're talking about its length and not the segment itself. And now what's its change? So this is its change in y.

And then the change in x, going from P to R, change in x is segment QR, the length of segment QR. And so we also know that these slopes are going to be the same because the slope of a line is constant everywhere. We also know that because these are similar triangles, if you take the ratio of the corresponding sides, you're going to have the same ratio.

So what we know is that this slope right over here needs to be equal to this slope right over here. Or we could say AB over BC should be equal to PQ over QR.

Now, which of these says that? So this one has PQ over QR and it has AB over AC, not BC, so I'll rule that one out. This one has PQ over QR; it has AB over BC. I'm liking this one.

Let's see. This one has QR over PQ; that's not the slope of this one over here, and so I'm not going to like that one. And this one also says QR over PQ and has AB over AC, so I don't like that one either.

So we're done. It's this choice B.

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