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Pablo and Leila are standing on riverbank; 170 meters apam; points and B respectively: (See the figure below ) They want t0 know the distancc from Leilu t0 house lo...

Question

Pablo and Leila are standing on riverbank; 170 meters apam; points and B respectively: (See the figure below ) They want t0 know the distancc from Leilu t0 house located Jcross the river at point C. Pablo measures angle 4 (angle BAC) to be 449, and Leila measures angle B (angle ABC) to be 63 What the distance from Lella t0 the house? Round your answer to the nearest tenth of meter170 mmeters

Pablo and Leila are standing on riverbank; 170 meters apam; points and B respectively: (See the figure below ) They want t0 know the distancc from Leilu t0 house located Jcross the river at point C. Pablo measures angle 4 (angle BAC) to be 449, and Leila measures angle B (angle ABC) to be 63 What the distance from Lella t0 the house? Round your answer to the nearest tenth of meter 170 m meters



Answers

Distance Across a River To find the distance across a river, a
surveyor chooses points $A$ and $B,$ which are 200 ft apart on
one side of the river (see the figure). She then chooses a ref-
erence point $C$ on the opposite side of the river and finds that
$\angle B A C \approx 82^{\circ}$ and $\angle A B C \approx 52^{\circ} .$ Approximate the distance
from $A$ to $C .$

Ah, a surveyor is looking at a river, and he has a point. A. So we have. If River here and he is here at point A and there's a point across the river, which is now he is wondering what is the distance between Point A and Point B in order to determine that without crossing the river she is going to use. He's going to use another point here on the same side that he labels point see. And so they're going to measure from Point A to Point B, and that is the call for 240 feet. Now you see, we can catch off perhaps our triangle. So we're wondering again, What is this like from a to B Now? Because the opposite angle is labeled seat. We're going label that unknown side as lower case. She's Casey that were wanting Got the distance from Hated me eso. We've talked about this 1st 1 and labeled that so the angle across from A to B so a TV show is here, and the angle cross from that, which is angle C is equal to 62 degrees. Angle across from sea to bees, feed a thieve angle across from that single A that is equal to 55 degrees and then find the distance from A to B. We label that as R. C. So in order to use the law of science to solve for this, we need that third angle measure because half side like feels here. This if I'd like way need angle here. Triangle consists of 180 degrees. So in order to go to find that third angle should do 180 minus 60 to minus 5580 my five minus soon equals 63 so that is ankle be 63 degrees. So in order to find this length, we used lost signs, which says that we can have five blank be over spine of ankle be on. This will be equal to decide, like see over side angle to put numbers in silence. Busy to 40 Angle B is equal to 63 degrees equals two side like see, we don't know what it is. It does a very bold over sign of angle sea just 62 degrees. Now we need to solve for C, so we will more supply both sides, but Science 62 cancels these out. So we have 140 time sign of 62. Over 63 is equal to startling. See, so far costlier and pictures in degree mode. So we have 240. Multiply by sign, uh, 62 divided by a fine of 53. That equals 237. Quite So. Watch around this to a whole number. Um uh You look at this age, it's about five. We round up, we'll have 238 feet sidelight.

Were given some points on a lake and were asked to figure out a certain distance. First, we're told that A and B are on opposite sides and that point C is 97 meters from a. So this distance right here is 97 meters. We are also told that the measure of the angle B A C is 100 in one degree. So if I draw the triangle B A c, the angle at B A C that's here is 101 degrees. I'm also told that the angle a CB, which would be here, has the size of 53 degrees. What I am trying to find is the distance from a Toby. Well, that distance from A to B is going to beat up here. Now I am gonna have to figure out this angle in order to properly use the law signs. Three angles in the triangle we know add up to 180. So if I take away the two that I already know, I'm left with 1/3 angle of 26 degrees. Now it's usual off science Side X has angle 53 open up to it to the first ratio is X over. Signed for the threat. The 97 meter distance has a 26 degree angle open up to it, so that's going to equal 97 over the sign of 26 degrees by multiply. Signed 53 to get X. By itself, I have 97 times the sign of 53 degrees over the sign of 26 degrees that will give me a total of 170 seven meters.

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.

So the question as justifying the distance between A and B, which our label as X so they provide us the following information. The distance between A and C is 97 meters, the angle C A B. There's 100 and one degrees, and the angle A C B is 53 degrees. So in order to solve this, we need to use the law of science. And to do that, we just need to find the value of this angle right here. Angle. So in order to do that, we just have to use the fact that all the angles and the triangle add up to 180 degrees and so tracked the value of the other two angles. So you get 1 80 minus 101 minus 53 and when you put this into your calculator, you'll get that this angle equals 26 degrees. Now we can use this law of signs to find X, so we'll use the fact that the sign of 26 is across from the only known side we have. His value is 97 and this will equal the sign of 53 which is across from X. So we just divide this my ex. And now we just have to put this in terms of x o X equals 97 times the sign of 53 all divided by the sine of 26. And when you plug this into your calculator, you'll get that X is approximately 176 meters.


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