how to deal with this?

A

allegansveritatem

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I can't even get out of the gate with this. I came upon it toward the end of my daily math study period so part of my problem might be brain fog...but I confess I don't even know where to take the first bite on this one:
rhom.PNG
 
It doesn't say any lines are parallel, so I doubt we can identify similar triangles. I'd apply the Law of Cosines to triangles BCD and BCE to find angles ABC and ACB, then focus on triangle ABC to answer the questions.
 
Dr.Peterson said:
It doesn't say any lines are parallel, so I doubt we can identify similar triangles. I'd apply the Law of Cosines to triangles BCD and BCE to find angles ABC and ACB, then focus on triangle ABC to answer the questions.
Click to expand...
Yes, you are correct about there not being parallel lines. I was mistaken about similar triangles.
You however are suggesting to find angles but the problems stated not to use angles (that is why I assumed similar triangles)
 
Dr.Peterson said:
It doesn't say any lines are parallel, so I doubt we can identify similar triangles. I'd apply the Law of Cosines to triangles BCD and BCE to find angles ABC and ACB, then focus on triangle ABC to answer the questions.
Click to expand...
I was thinking about using the law of cosines--for one thing, this problem comes at the end of a section where the law of cosines is introduced--but I was put off by the statement that we can find what we are looking for without measuring angles and I guess I took that phrase "without measuring angles" to mean "without knowing angles" but of course, that would make the solution impossible. Yes, the nest of triangles in that parallelogram are where the bird of resolution is lurking. I will take that tack and post my results tomorrow. Thanks
 
Jomo said:
Yes, you are correct about there not being parallel lines. I was mistaken about similar triangles.
You however are suggesting to find angles but the problems stated not to use angles (that is why I assumed similar triangles)
Click to expand...
yes, that business about not needing to measure angles put me off too, but I think the author didn't not express himself fully...he should h ave made it clear that measuring the angles is not necessary provided you know certain dimensions that will allow you to find trigonometrically what the relevant angles are. How else can we solve this without knowing at least a few angles?
 
Jomo said:
Yes, you are correct about there not being parallel lines. I was mistaken about similar triangles.
You however are suggesting to find angles but the problems stated not to use angles (that is why I assumed similar triangles)
Click to expand...
Read carefully. It distinctly says, "without measuring angles", which is entirely different from never using an angle in your work. In the problem, they measured some distances, but no angles, and that's all they mean.
 
Dr.Peterson said:
Read carefully. It distinctly says, "without measuring angles", which is entirely different from never using an angle in your work. In the problem, they measured some distances, but no angles, and that's all they mean.
Click to expand...
right.
As an aside I want to say that I woke this morning thinking of this problem and it dawned on me I had made a mistake yesterday when I referred to the figure on the near shore in the diagram as a parallelogram. No time to be sure right now but I think I should have said trapezoid.
 
allegansveritatem said:
right.
As an aside I want to say that I woke this morning thinking of this problem and it dawned on me I had made a mistake yesterday when I referred to the figure on the near shore in the diagram as a parallelogram. No time to be sure right now but I think I should have said trapezoid.
Click to expand...
No, there's no trapezoid either; nothing is stated to be parallel, though the picture looks as if it could be. It's just a quadrilateral.

Here is a scale drawing:

1593440037221.png

Just do the trig! Don't overthink it.
 
allegansveritatem said:
right.
As an aside I want to say that I woke this morning thinking of this problem and it dawned on me I had made a mistake yesterday when I referred to the figure on the near shore in the diagram as a parallelogram. No time to be sure right now but I think I should have said trapezoid.
Click to expand...
It is not a trapezoid either. BC and DE are not given to be parallel. So, as given, BCED is just a quadrilateral.
 
Dr.Peterson said:
No, there's no trapezoid either; nothing is stated to be parallel, though the picture looks as if it could be. It's just a quadrilateral.

Here is a scale drawing:

View attachment 20070

Just do the trig! Don't overthink it.
Click to expand...
I am so dumb that I didn't know that part of the definition of a trapezoid is that two sides must be parallel. Anyway, I worked on this two hours today and after a few false starts I realized what I needed to do, namely find some angles with the law of cosines and then use the law of sines to get the dimensions on the transriver triangle. Here is what I did:(my results are given the lie by my diagram but that isn't necessarily a deal breaker here:
rhom2.PNG

I did not have the time to check this but I put it out there anyway. I will check it tomorrow.
 
There may be a rounding error in there, but it looks pretty good. Here is my drawing with labels added:
1593461262019.png
 
I should have paid more attention to scale in my drawing...it would have saved me a little from doubting my results. But I took my cue from the book and threw the thing together any old way.
I must say I spent a long long time fooling with this problem without having anything other than tactical objectives; at one point I had every angle (except the 4 in the center) in the trapezium--a new word for me--worked out , all eight of them, with no strategy for using them. Then it dawned on me that I could find the two angles at the base of the river triangle and get the third angle from those and then use the law of sines. It should have been obvious from the start, but somehow my tuner was locked onto the wrong station.
 
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On important part of strategy is to give some thought early to the "end game", thinking about what might be the last few steps, in order to direct your tactical thinking. Since you are looking for AB and AC, you should expect to work in triangle ABC at the end; and since all the given data are outside the triangle, you might see that the angles at B and C make a good transition. So that would become my initial goal.

By the say, I suggested this strategy in post #3.
 
Dr.Peterson said:
On important part of strategy is to give some thought early to the "end game", thinking about what might be the last few steps, in order to direct your tactical thinking. Since you are looking for AB and AC, you should expect to work in triangle ABC at the end; and since all the given data are outside the triangle, you might see that the angles at B and C make a good transition. So that would become my initial goal.

By the say, I suggested this strategy in post #3.
Click to expand...
I could see that BC was the interface between the partially known and the unknown...but somehow I was fixed on the idea that I needed to know the sides of ABC and then I might be able to find the angle at A and use the law of cosines. I had law of cosines on the brain even though I knew that it was common procedure to use both law of cosines and law of sines together. So I guess I had a plan of some kind after all. It just wasn't a good one. I think it was Mark Twain who said:"It's not what you don't know that hurts you, it's what you know that just ain't so."
 
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