This problem is kind of a difficult one. And I hope, I hope I did it right. I think so. Answers make sense. So where we want to figure out this, what are they actually asked us? The distance is here? Um So the distance across the river shown in the figure can be found without measuring angles. So we we two points BNC on opposite shore. Okay, so here in here are selected points um and line A B and a C. Wait a minute, B and C. So these total points here are given here and then we extend those throughout here. Um and the line segments are extended as shown. So we have a A here. Let me extend them here and extended out here. And then we have these crosses here that we measure. So we can measure this, can measure this, we can measure this, we can measure this this and we can also measure that. But they didn't give us that information and what that information. We can figure out what this like this and what this length is now. Um basically what you have to do that, you have to make sure you cite to something you know here. So you can site along and keep this these site something comin over here. Like if there's a tree or something over here, So they tell us b. 104 80 584 ft, b. d. is 100 2 ft. E. There's 218 ft. 236 ft and c. e. is 80 ft. So what you need to do, what I decided to do is I was going to figure out if we can figure out what, let's see here, You can look at this triangle here and figure out this angle And then this angle here is 180- that. And then we can look at let's see here this triangle here And then find this angle and then this angle is 180- that. So We have I called 31 the angle the B C. So this angle here and B C E. This angle here, I called David too. So we know C D squared equals B C squared plus B D squared minus to B C. Times B D. Times co signed a 3 to 1. And we know all these links so we can solve the theater one. And that turns out to be 100 and 100 and seven degrees. Now we can let's see here, we can get this other angle data to B squared equals C E squared plus B c squared minus two cbc times co sign of data to You know everything in here except for data to so we can get that and that's 103°.. So that means that this angle here between here and here is the angle C. Or actually I have C. B. A. This one here is 73 degrees and the angle B C. A. This one here is 70 six degrees. All right. Um So then let's see here, we know then that this angle here. So we know this angle, we know this angle, we can look at this triangle here. So the total angles have to add up to 180°.. So, you know this angle is 35°.. Now let's see here. Now we can use the law of sines because we know we know all the angles had this triangle and we know this length here. So we can get this lengthen this length. Youth in with the Law of science. and so that says that a. b. 100 and 54 ft And AC. 138 ft. Um And again this isn't really drawn to scale, looks like hopefully these answers are correct but it looks like is longer and the way this is drawn anyway. But these these these it also looks weird because you know, in the they give what they give us the B. D. Is you know, quite a somewhat longer than ce but at wrecked it looks like ce is longer in the picture. So anyway, it's obviously not really drawn to scale. So then they ask us about the shortest distance across the room. So that would be basically I took this triangle here and just kind of looked down on it and through this triangle here. So we have A. B. C. And we know say that three and we know theatre four and we know 35. And so the shortest distance across to point B. A. Would be if we weren't perpendicular. So if we weren't perpendicular to this, that would be the shortest. So what we can do then is no we know then that A. F. Equals A. C. Times a sign of this angle here Because this is 90°.. So that means a. 380 338 .6 ft. So that would be the shortest distance across the river um to point A. And then we could figure out what this distance is too. So we can know where to way to launch from. I guess. So I think that's just a lot of a lot of geometry here. I got I did all the calculations right and the numbers seem reasonable. So I think everything looks good.