Applying similar triangles

0:00- [Instructor] If we're told a six meter high building casts a 10-meter shadow on level ground, a person 165 centimeters tall wants to stand in the shade as far away from the building as possible. What is this distance? Pause this video. Have a go at this before we do this together.

0:21Alright, so let's first think about the building. So a six meter high building, so maybe this is the building right over here. Six meters, just like that. So this distance right over here is six meters. I really just needed to draw one wall of the building. So that's six meters, cast a 10 meter shadow on level ground.

0:4410 meter shadow on level ground, so that could look something like this. So 10 meters looks something like this. So this distance right over here is 10 meters. We're assuming that the sunlight is coming like this. And actually let me draw the light, so the light is coming in and the shadow cast.

1:06And so anything, oh, let me draw a little bit better than that. So connect those dots. There you go. So one way to think about it is there's, all the light is coming up here. It's not being blocked by the building. And what we're thinking about is a person that's 165 centimeters tall, and how far can they walk?

1:25Well, they can walk until their head starts to poke through and get some of that light. So 165 centimeters, that's 1.65 meters. And actually let me just do everything in meters so that we don't get ourselves confused. So 1.65 meters, and I'm just eyeballing it.

1:44This isn't drawn to scale. Might put us right about here. So one point, so this distance is 1.65 meters. That's the person's height. So there's their head, their arms, their feet, okay, you get the idea. So the question is, is what is this distance? What they want to go as far away from the building as possible.

2:07If they go any further, their head's going to dip into the sun. They're not gonna be fully in the shadow. If they get any closer, they're not as far away as possible. So that distance is X. So you might be thinking, hey, these, it looks like we might be dealing with some form of a similar triangle here, and you would be correct.

2:27And the way to think about that is, we have a lot of, there's many ways you could approach it, but you could think, all right, I have this larger triangle. I have this larger triangle right over here. This is a right angle, and let's call, let's just, that's some angle right over there.

2:45And so let me draw that separately, so I have that larger triangle. And so I have that angle, and I have that right angle. And then I also have, let me do this in another color. I also have this smaller triangle where that person is one side of it. And really you could think of it as the person's shadow is the other side of it.

3:04So I also have this smaller triangle, but this is also gonna be a right angle. So the smaller triangle looks like this. It looks like this. You have a right angle, and they both share this angle right over here. So that's common. Well, we know if you have two triangles that have two angles in common, they're going to be similar.

3:21In fact, you also know that the third angle is also going to be in common 'cause you could figure that out based on the other two. So when you have all of the angles of two triangles being congruent or having the same measure, you have similar triangles, and that means that the corresponding sides are going to be proportional.

3:39So one way to think about it is, before we think about X, let's think about, I'll do a different letter here. Let's call this distance right over here, I don't know, let's call that Y. So if that is, if we could figure out Y, we just subtract Y from 10 and we will figure out X.

3:56So we already know that this height right over here is 1.65 meters. We know that this distance right over here in the big triangle is 10 meters. Actually, I'll just assume everything's in meters. And we know that the height of building is six meters. Six meters.

4:15So one way to think about it, we could say Y over 1.65, Y over 1.65, 1.65 is going to be equal to, make sure I get the corresponding side. So the long side here is 10, it's going to be equal to 10 over six. I probably shouldn't have reused the purple, but you get that's the six right over there.

4:39And then to solve for Y, we just multiply both sides by 1.65, 1.65 times that side, and 1.65 times that side, that cancels. And I get Y is equal to, let's see, 10 times 1.65 is 16.5. So it's going to be 16.5 divided by six. And I can get a calculator out for that.

5:0316.5 divided by six is equal to that. Now you might say, oh, is the answer 2.75 meters? Well, we're close but not quite. Remember Y is 2.75, but X, which is how far you can walk away from the building, is going to be 10 minus 2.75. So if you were to subtract that from 10, which you could probably do it in your head, you get 7.25.

5:27So once again, we get Y. This distance right over here is 2.75. But then if you want X, we get X, we do x in that same red color. X is equal to, is equal to 10 meters minus 2.75 meters, which is equal to 7.25 meters. So that's how far the person can walk away from the building.