Multi-step word problem with Pythagorean theorem | Geometry | Khan Academy

We're told that Laney 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 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.

What they're going to do, what Laney's doing, is she is stringing up these lights. So that's this yellow right over here. 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. It would look 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 that's six meters away.

So the yellow right over here, that is the light. And so we need to figure out how many total lights are on the string. The way I would tackle it is first off, first of all, I want to figure out how long the total string is.

To figure that out, I just need to figure out, okay, it's going to be 2.5 plus whatever the hypotenuse 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.

So we have to figure out the length of this hypotenuse. If we know that plus this 2.5 meters, then we know how long the entire string of lights is. 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. How do we figure out the hypotenuse here? Well, of course, we would use the Pythagorean theorem. So let's call this, let's call this h for hypotenuse. We know that the hypotenuse squared is equal to 2.5 squared plus 6 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 hypotenuse is going to be equal to the square root of 42.25. 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 figured out the hypotenuse, let me just multiply that by 10." In fact, my brain almost did that just now, but we got to realize that the entire string is the hypotenuse plus this 2.5.

So the whole string length, let me write this way: string length is equal to 2.5 plus the square root of 42.25. Then we would just multiply that times 10 to get the total number of lights.

So now let's actually get the calculator out. We have 42.25, and then we were to take the square root of that, which gets us to 6.5. Then we add the other 2.5, plus 2.5 equals 9. That's the total length of the string.

So the total length of the string is 9 meters, and there are 10 lights per meter. So the number of lights, the number of lights is equal to 9 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 fence posts, that maybe you could argue that there's one extra light in there. But for the sake of this, I kind of view this as on average there are 10 lights per meter. And so if it's a 9 meter string, 9 times 10 is 90 lights.