Northernmost Point of a Route: Finding Distance and Latitude from North Pole

[MrsTesla]

Homework Statement

Find the distance between Helsinki and Seattle along the shortest route. Where is the northermost point of the route, and what is its distance from the North Pole? The longitude of Helsinki is 25ºE and latitude 60º; the longitude of Seattle is 122ºW and latitude 48º. Assume the radius of the Earth is 6370km

Homework Equations

3. The Attempt at a Solution [/B]
I already solved the first part of the problem (the distance), which is 7640km, using spherical trigonometry, but I have no idea how to solve the second part of the problem.

 

[Orodruin]

Please show how you solved the first part. It will help in judging what you have already done and give you better advice on how to continue.

 

[MrsTesla]

This is what I did. The final result is correct according to the book, but I'm not sure I have all steps right.
I'm sorry if the quality of the image is not the best.

 

[Orodruin]

So let me ask you the following question: If you had a triangle in Euclidean space and wanted to find the closest distance from one of the sides to the opposite corner, what would you do?

 

[MrsTesla]

I would draw a straight line perpendicular to the chosen side passing through the opposite corner. This would make a right triangle and from that, knowing the angles and sides, I would be able to calculate that distance.

 

[Orodruin]

So what stops you from doing that in spherical geometry?

 

[MrsTesla]

Okay, I got it. I have no idea why I didn't think about that.
Thank you

 

[Orodruin]

Would you mind posting your solution for the benefit of anyone who might come across this thread? Also, according to forum rules, posting alternative solutions is not allowed until the OP has shown that he/she has solved the problem.

 

[MrsTesla]

Here's my solution to the problem.