Wie lang ist die rote Strecke? – Mathe RÄTSEL Geometrie

Hello everyone! Today I have brought you this exercise in which we are supposed to find out how long this red line is. We are given that our big triangle here, down here in the corner, has a right angle. Then this side is given here and this side down here and this angle here with 30 degrees. These are all the information we get and now we have to see how we can find out how long this red line is. You are welcome to try it out for yourself first, how you would proceed here. I'll show you a way to proceed here. Maybe we give our side a name and call it x, that's what we want to find out. And our line is actually just in the middle of this triangle. But if we make a small triangle down here, then x is just one side of our triangle. And it's very practical that our side is basically just in a triangle. This triangle is known for one side down here and an angle. And the more we get out of our triangle, the easier it is to calculate this missing side. Maybe we'll take care of the angle first. I'll just call this angle alpha and we'll see how to calculate it here. This angle alpha also comes in our big triangle up here. This is just this angle down here. And the big triangle is a right-angled triangle. Two sides are known from it, which is very practical, because then we can calculate this missing angle with sine, cosine or tangent. Now we just have to see which one we choose. For sine and cosine, we need the hypotenuse. Is it given here? This is the side that lies opposite the right angle. So this is this side here. It's not given. You could calculate it, but you don't have to. But with that, sine and cosine are out because they use the hypotenuse. But not the tangent. And that's why we take the tangent of our chosen angle and use the other two sides for it. The tangent of alpha is defined as the antithesis divided by the antithesis. The antithesis is always the side that lies opposite my angle, which I'm looking at right now. So let's look at alpha here. That's why we go to alpha and see which side is opposite. It's given here with 15, so we can use that for the antithesis. The antithesis is the side that lies opposite my angle, which I'm looking at right now. And not the hypotenuse. So we've already identified the hypotenuse. It's out there. The antithesis is on the other side. And it's given here with 30. We can enter that here. And then we have a small equation here where only alpha occurs. And we can solve that for alpha. Only the tangent has to disappear here. So let's apply the inverse function of the tangent. So the arcus tangent or tangent to the power of minus 1. This dissolves the tangent and only alpha remains on the left side. But we also have to apply it to the right. So tangent to the power of minus 1 from the whole side, as it was there, we can simply enter here. You could also shorten the break, but you can also leave it as it is. You type the whole thing into a calculator anyway. And alpha, if we round that to two decimal places, we get a value of 26.57 degrees. And then we've already found something. And I've entered that here. So that means we've given two angles from our little triangle down here. That means for us this third angle, we can definitely calculate it. Because in a triangle, the inner angle sum is 180 degrees. So if we add up all three angles, we get 180 degrees. That means for our chosen angle beta, if we subtract the other two angles from 180 degrees, then we can calculate how big our chosen angle beta is. And if we do that, we get a value of 123.43 degrees. And with that, we've calculated all three angles in our little triangle here. What does that do now? Well, we'll have to see. We don't have a right-angled triangle down here, so unfortunately no 90 degrees anywhere. That means sine, cosine or tangent are out. But in such general triangles, for example, the sine or cosine set also applies. And we could apply the sine set here, because it uses two sides and two angles. And we've given one side here. And it's very important to look at the opposite angle to this side of the sine set. And one side is chosen, but we have the opposite angle for that. And that's perfect for the sine set. It says, take one side of your triangle, divide it by the sine of the opposite angle, in our case our 26.57 degrees. And then it's the same as if you took another side of your triangle, the 30, and divide that by the sine of the opposite angle, so sine of our 123.43 degrees. And with that, we have a equation here that doesn't look very nice, but it's not bad at all. Because there's only x in there as unknown. Everything else you can just type into a calculator and calculate. There are only numbers and sine, but that's what our calculator does for us. So if we want to solve for x, this break has to go. So we just have to multiply the equation by the sine of this angle. And with that, we can calculate x by typing in this break. 30 divided by the sine of this huge angle. And then don't forget that we just did a transformation. So take the sine of the smaller angle again. And if you type all of that into your calculator and don't get confused, you'll get a value for our red line of 16.08. And since we were using meters here, that's also a unit of meters. I hope you liked it. I'm curious to see which ways you all found. Feel free to write them in the comments. Otherwise, have a wonderful day and we'll see you in the next video. Take care!