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Multi-step word problem with Pythagorean theorem

CCSS.Math:

Video transcript

- [Instructor] We're told that Lainey runs a string of lights from the ground straight up to a door frame that is 2.5 meters tall. Then they run the rest of the string in a straight line to a point on the ground that is six meters from the base of the door frame. There are 10 lights per meter of the string. How many total lights are on the string? So I'll pause this video and see if you can work this out. All right, now let's work through this together. I think this one warrants some type of a diagram. So let me draw this door frame that looks like this. And this door frame is 2.5 meters tall. So that's its height right over there. And what they're going to do, what Lainey's doing is she is string up these lights, so that's this yellow right over here. So it goes up to the top of that door frame and then they run the rest of the light in a straight line to a point on the ground that is six meters from the base of the door frame. So let me show a point that is six meters from the base of the door frame. So it looks something like, maybe like that. So this distance right over here is six meters. So they run the rest of the light from the top of the door frame to that point to that six meters away. So the yellow right over here, that is the light. And so we had to figure out how many total lights are on the string. So the way I would tackle it is first of all, I wanna figure out how long is the total string. And to figure that out, I just need to figure out, okay, it's gonna be 2.5 plus whatever the hypothenuse is of this right triangle. I think it's safe to assume that this is a standard house where door frames are at a right angle to the floor. And so we have to figure out the length of this hypothenuse and if we know that, plus this 2.5 meters, then we know how long the entire string of lights are. And then we just have to really multiply it by 10 because there's 10 lights per meter of the string. So let's do that. So how do we figure out the hypothenuse here? Well, of course, we would use Pythagorean theorem. So let's call this, and also call this h for hypothenuse. We know that the hypothenuse squared is equal to 2.5 squared plus six squared. So this is going to be equal to 6.25 plus 36, which is equal to 42.25 or we could say that the hypothenuse is going to be equal to the square root of 42.25. And I could get my calculator out at this point, but I'll actually just keep using this expression to figure out the total number of lights. So what's the total length of the string? We have to be careful here. A lot of folks would say, "Oh, I figure out the hypothenuse, let me just multiply it that by 10." In fact, my brain almost did that just now. But we gotta realize that the entire string is the hypothenuse plus this 2.5. So the whole string length, Write this way. String length is equal to 2.5 plus the square root of 42.25. And then we would just multiply it that time 10 to get the total number of lights. So now let's actually get the calculator out. So we have 42.25 and then we would've take the square root of that. Gets us to 6.5 and then we add the other 2.5, plus 2.5 equals that. That's the total length of string. So the total length of string is nine meters and there're 10 lights per meter. So the number of lights, number of lights is equal to nine total meters of string times 10 lights per meter, which would give us 90 lights. Now some of you might debate, if you think really deeply about it, is if you have a light right at the beginning, if these were kind of set up like a fence post, then maybe you can argue that there's one extra light in there, but for the sake of this, I kind of view this as, on average there're 10 lights per meter and so if it's a nine meter string, nine times 10 is 90 lights.