Applying the triangle proportionality theorem

0:00- [Instructor] We're told, in San Antonio, Texas, Sinclair Road, Mary Helen Drive, Irene Drive, and Theis Drive run parallel to each other. We have a map here of part of San Antonio, Texas, and we can see that, we have Sinclair Road, Mary Helen Drive, Irene Drive and Theis Drive are parallel to each other.

0:19They intersect South WW White Road, this is South WW White Road, and Roland Ave. right over there, forming a triangle since White and Roland intersect. Yep, they intersect right over there. So I can see the triangles that each of these parallel roads form with those other two roads.

0:37Along South WW White Road, it is 0.17 kilometers from Sinclair Road to Mary Helen Drive. So from Sinclair Road to Mary Helen Drive. So this distance right over there is 0.17 kilometers. So let me underline where I'm getting this information from and then they tell us, and it's 0.56 kilometers from Irene Drive to Theis Drive.

1:11So once again, along WW White, so Irene to Theis Drive. So this distance right over here, that is 0.56 kilometers. All right, a lot of information here. Then they tell us, along Roland Road, it is 0.2 kilometers from Sinclair to Mary Helen Drive. So along Roland Road, it is 0.2 from Sinclair to Mary Helen Drive.

1:39So that's this distance right over here, 0.2 kilometers. What is the approximate distance, in kilometers, along Roland Road, Roland Road between Irene and Theis Drive? So they are asking us between Irene and Theis Drive. So this is what we wanna figure out. So why don't you pause this video and see if you can figure this out by yourself before we do this together.

2:07All right, now let's do this together. So the key here is that Theis, Irene, Mary Helen, and Sinclair are all parallel to each other and they all intersect the same two roads, this WW White Road and Roland Road. And each of them forms triangles, and actually, they're forming similar triangles and that comes out of the Triangle Proportionality Theorem, which goes even further than that.

2:31That tells us that if a line intersects two sides of a triangle and is parallel to another side or another line, then the line divides the two sides proportionally. And this is true no matter how many parallel lines you have. So one way to think about it is, these are all parallel lines.

2:52And so the respective sides of the different triangles, the lengths in between them are going to be proportional. So another way we could say that is the ratio of X to 0.56 kilometers is going to be the same as the ratio of 0.2 to 0.17. Let me write that. So the ratio of X to, I'll do the same colors, 0.56, we'll just assume these are all in kilometers so I don't have to keep rewriting the units, that's the same as 0.2, 0.2 over, over 0.17.

3:28And so if we wanna solve for X, we just multiply both sides times 0.56. So 0.56 times that side and 0.56 times that side. That will cancel. And we're going to get X is equal to, I'll get out a calculator for this one, 0.2 times 0.56 divided by 0.17 is equal to zero point, and I'm gonna round to the nearest hundredths, because that looks like reasonable, considering a lot of these other things are in hundredths.

4:05Some could argue to the nearest tenths, 'cause they only told us that much, but I'm assuming we're going to the hundredths and this was just exactly 0.2. Maybe it's 0.20. So let's say 0.66 kilometers. So X is equal to 0.66 kilometers, and we are done. It's always good to look and make sure, oh, yeah, it looks about right too.

4:27Obviously, you can't do it just based on looks, but you can kind of do a reasonable test. If you saw six kilometers here or if you saw 0.06 kilometers, you're like, "Okay, that doesn't seem right."