Please help with this pyramid problem
- Thread starter s garg
- Start date
Dr.Peterson
Elite Member
- Joined
- Nov 12, 2017
- Messages
- 16,604
First, it appears that you either didn't read our submission guidelines, or chose not to cooperate. Please do so, by telling us the context of the problem, and where you need help.
Second, I wonder it you omitted some information or a picture. Some interpretation is needed. If "all the altitudes are equal" refers to the altitudes of the faces, then they would have to be equilateral. It must therefore be taken to mean the altitude from each vertex to the opposite face.
The next step is for you to tell us what you know about pyramids (tetrahedra), and what thoughts you have for solving this. I presume you've at least drawn a picture and checked for formulas.
Second, I wonder it you omitted some information or a picture. Some interpretation is needed. If "all the altitudes are equal" refers to the altitudes of the faces, then they would have to be equilateral. It must therefore be taken to mean the altitude from each vertex to the opposite face.
The next step is for you to tell us what you know about pyramids (tetrahedra), and what thoughts you have for solving this. I presume you've at least drawn a picture and checked for formulas.
I am really sorry sir i am new to this site, my friend told me about the awesome work you guys do to help. and honestly i am preparing for an exam and found this problem in a book. It had just this no diagrams nothing else. Yes i have a picture i made and i thought of solving it using some trigonometry but, i have never really done a question dealing with pyramids before so i know little.
Dr.Peterson
Elite Member
- Joined
- Nov 12, 2017
- Messages
- 16,604
Let me fill in what you haven't said. You are preparing for an exam which is not for a course you are taking but ... for what? (For college entrance? For employment? Of what kind?) You found this problem in a book ... of sample problems for that exam? (Or just general difficult problems?) Possibly knowing what the exam is for, and what the book says it covers would help us see how you might be expected to solve the problem. Finally, you know some trigonometry ... but how much? Does the exam expect you to know something beyond that?
I will now admit that I have not yet tried solving the problem, because I don't know what "tools" to try using, and because (with no context) it is a big challenge. I'll give it a try as I have time, and maybe someone else here will have a better idea than I do, but getting the information I've asked for is the first step.
Thank you sir for trying,
The exam is IIT JEE and it is for college entrance and for a course i am going to take. i know most of the stuff in trigonometry including the inverse trigo. The book is for the sample problems of the exam. I just do not have any idea to solve this one as it is the first problem i have faced dealing with pyramids.
Dr.Peterson
Elite Member
- Joined
- Nov 12, 2017
- Messages
- 16,604
Thanks. I'll try to find time to look into this problem.
I have to say that I have seen sample questions for IIT JEE exams in the past, and wondered if they were really representative of the exam, because they seemed far too difficult for that context, as this one does. (It's possible that some questions are intended to reveal mathematical geniuses who are far above their expected level, and most of you can get by without being able to do them.)
I have to say that I have seen sample questions for IIT JEE exams in the past, and wondered if they were really representative of the exam, because they seemed far too difficult for that context, as this one does. (It's possible that some questions are intended to reveal mathematical geniuses who are far above their expected level, and most of you can get by without being able to do them.)
Dr.Peterson
Elite Member
- Joined
- Nov 12, 2017
- Messages
- 16,604
Here is one idea for you to pursue. (Remember, the goal is for you to learn to solve problems yourself, not just to see someone else do it; I haven't tried carrying this out.)
Try using coordinate geometry. I suggest putting K at (0, 0, 0), L at (14, 0, 0) on the x-axis, N at (a, b, 0) on the xy-plane, and M at (c, d, e). This requires us to solve for 5 variables.
Then the fact that NK = 17 gives us the equation a^2 + b^2 = 17^2.
The altitude to M is e, so you can write three more equations stating that the altitudes to K, L, and N are all equal to e, using the formula for the distance from a point to a plane. (If you don't know that, look it up or derive it; one way to do this uses vectors.)
The fifth equation we need will come from the fact that ∠MLK = 60°. One way to write this equation is to use the dot product of vectors.
If you don't know vectors, there are other ways. I had hoped you might have mentioned knowing about them.
Try using coordinate geometry. I suggest putting K at (0, 0, 0), L at (14, 0, 0) on the x-axis, N at (a, b, 0) on the xy-plane, and M at (c, d, e). This requires us to solve for 5 variables.
Then the fact that NK = 17 gives us the equation a^2 + b^2 = 17^2.
The altitude to M is e, so you can write three more equations stating that the altitudes to K, L, and N are all equal to e, using the formula for the distance from a point to a plane. (If you don't know that, look it up or derive it; one way to do this uses vectors.)
The fifth equation we need will come from the fact that ∠MLK = 60°. One way to write this equation is to use the dot product of vectors.
If you don't know vectors, there are other ways. I had hoped you might have mentioned knowing about them.