Project Survey: A surveyor

Question

Project Survey: A surveyor at point A on the south side of a canyon needs to find the distance between the tree T and the rock formation R on the canyon’s north side ( , see figure). To make this possible, she takes a bearing from point A, and finds that point R is 70º East of North, and point T is 30º East of North. She then measures a straight line distance of 30 m to a point D on her side of the canyon. From there she finds that point T is 68º West of North, and point R is 9º West of North. Assume the dashed lines intersect at a point C. Use this information to answer the questions that follow.

Answer 1

Dev, I have also outline a solution to this problem by posting a jpeg document on the web.  A direct link to the webpage is

 

http://i.imgur.com/1HV4XP1.jpg

 

Answer 2

Dev,

 

Is the question to find the distance between the tree and the rock formation? You never made this clear, so I will answer this hoping it is what you wanted.

 

Consider triangle ADT. You know the length AD = 30 m, and the two angles TAD = 60° and TDA = 22° are easily found. You can then find the third angle, ATD = 180° - 60° - 22° = 98°. You can now use the Law of Sines to find the other two sides, AT and DT.

 

Similarly, in triangle ADR, you know AD = 30 m, angle ADR = 81°, and angle RAD = 20°. You can find angle ARD to be 79°. You can then find the sides DR and AR using the Law of Sines again.

 

You stated in your question that the two diagonals intersect at a point C, but you did not state anything more about this, so I will assume no further information is given.

 

Consider the triangle ACD. You know AD = 30 m, angles CAD = 20° and angle CDA = 22°, from which you can find angle ACD = 138°. Use can use the Law of Sines again to find the sides AC and DC. By subtracting their lengths from AR and DT, respectively, you can find the lengths CT and CR.

 

Lastly, consider the triangle CTR. You know that angle TCR = angle CDA = 22° (vertical angles). You also now know the lengths CT and CR because they were just found in the last paragraph. So, to find the third side, RT (the distance between the tree and the rock formation), you can use the Law of Cosines in this triangle.

 

What I have given here is only an outline of a solution. I left the details and answers out because the main problem was the problem solving process and not just the answers.

 

I suggest you try to complete the problem based upon my extensive hints and follow up with me in another message if you have any more questions. This was obviously a very long and multi-step problem and I am just trying to get you started.

 

One more word of advice. When you post a very tough question like this, PLEASE be sure it is complete and no wording or notation is omitted. My "sketch" above assumed a few things and if I was wrong, I basically wasted your time and mine. Let me know if any of this helps you or if need further assistance with the problem. I am glad to help you finish this.