Finding the last corner of a square

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Homework Help Overview

The problem involves determining the coordinates of point C in a quadrilateral ABCD, given points A(-2|2|3), B(2|10|4), and D(5|-2|7), with the requirement that ABCD forms a square. Participants are exploring the relationships between the lengths of the sides and diagonals of the square.

Discussion Character

  • Mixed

Approaches and Questions Raised

  • Participants discuss measuring distances between points and question the validity of their calculations. There is exploration of vector addition and the implications of the order of points in defining the quadrilateral. Some participants express uncertainty about their understanding of vectors and how to apply them to find point C.

Discussion Status

The discussion is ongoing, with participants providing insights into vector manipulation and the geometric properties of squares. Some guidance has been offered regarding the relationship between the sides and diagonals, but there is no explicit consensus on the correct approach to finding point C.

Contextual Notes

Participants mention missing classes, which may contribute to gaps in understanding vector operations and geometric principles relevant to the problem.

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Homework Statement


Show that there is a C point to these points A(-2|2|3), B(2|10|4) and D(5|-2|7), so that the quadrangle ABCD a square is. Determine the coordinates of C.

The Attempt at a Solution


All I have done is measure AC, CD and BC, which are 9, 9 and 12.78. But how do I get the C point from this data?

Thanks in advance
 
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H4NS said:

Homework Statement


Show that there is a C point to these points A(-2|2|3), B(2|10|4) and D(5|-2|7), so that the quadrangle ABCD a square is. Determine the coordinates of C.

The Attempt at a Solution


All I have done is measure AC, CD and BC, which are 9, 9 and 12.78. But how do I get the C point from this data?

Thanks in advance

The length of AB = 9 by calculation and the length of AD = 9 by calculation, so AD and AB are adjacent sides, with BC a diagonal of length 12.78. That means AC must be a diagonal and its length can't be 9. Something is wrong with your figures.
 
LCKurtz said:
The length of AB = 9 by calculation and the length of AD = 9 by calculation, so AD and AB are adjacent sides, with BC a diagonal of length 12.78. That means AC must be a diagonal and its length can't be 9. Something is wrong with your figures.

Oh Sorry, I meant AC = 12.78, since it's a square, BC and CD must be 9 too.
 
Usually when you give a quadrilateral ABCD that would give the vertices in order as you go around the square. In this case with AB and AD adjacent sides, the order around the square can't be ABCD. Is that what you meant to have or do you still have typos? It might be good to state carefully exactly what you are given.

Have you studied vectors and how to manipulate them?
 
LCKurtz said:
Usually when you give a quadrilateral ABCD that would give the vertices in order as you go around the square. In this case with AB and AD adjacent sides, the order around the square can't be ABCD. Is that what you meant to have or do you still have typos? It might be good to state carefully exactly what you are given.

Have you studied vectors and how to manipulate them?
m21Ca.png

This is what I mean

I have also missed some classes, and that's why I require help from you
 
LCKurtz said:
Have you studied vectors and how to manipulate them?

H4NS said:
m21Ca.png

This is what I mean

I have also missed some classes, and that's why I require help from you

You didn't answer my question whether you have studied vectors. Think about what you would get if you add the vectors AB and AD. What direction would it point?
 
LCKurtz said:
You didn't answer my question whether you have studied vectors. Think about what you would get if you add the vectors AB and AD. What direction would it point?
AB+AD would point in the direction of AC, but I don't know those vektors, I only know the points ABD

I know how to perform certain operations with vectors, but as I said, I missed some classes, and probably the key to resolve this problem
 
H4NS said:
AB+AD would point in the direction of AC, but I don't know those vektors, I only know the points ABD

I know how to perform certain operations with vectors, but as I said, I missed some classes, and probably the key to resolve this problem

To get a vector between points P and Q you subtract the coordinates of P from the coordinates of Q to get the components of the vector PQ.
 
LCKurtz said:
To get a vector between points P and Q you subtract the coordinates of P from the coordinates of Q to get the components of the vector PQ.
Oh man, thanks a lot, then I suppose I have to do the following:

AB: (2-(-2)|10-2|4-3) AD: (5-(-2)|-2-2|7-3)

Then AC: (4+7|8-4|1+4)

Thus the C point is (11|4|5), right?

But this is somehow wrong because when doing calculating CD or BC I get don't get 9!

What am I doing wrong?
 
  • #10
H4NS said:
Oh man, thanks a lot, then I suppose I have to do the following:

AB: (2-(-2)|10-2|4-3) AD: (5-(-2)|-2-2|7-3)

Then AC: (4+7|8-4|1+4)

Thus the C point is (11|4|5), right?

But this is somehow wrong because when doing calculating CD or BC I get don't get 9!

What am I doing wrong?

No, that is not the point C. It is a vector pointing towards C. You have a vector in the right direction, but its length is whatever it came out. How long is it? How long does it need to be to fit the diagonal? How do you change its length?
 
  • #11
LCKurtz said:
No, that is not the point C. It is a vector pointing towards C. You have a vector in the right direction, but its length is whatever it came out. How long is it? How long does it need to be to fit the diagonal? How do you change its length?
It's length is as far as I know is |C|, 12.72, which is the measure of the diagonal
 
  • #12
H4NS said:
It's length is as far as I know is |C|, 12.72, which is the measure of the diagonal

OK, so it is the right length. Now, if you are at the point A, how do you use that vector with the coordinates of A to figure out the coordinates of C? And, by the way, it would be best to use exact values instead of decimal approximations if you want the exact coordinates of C.
 
  • #13
Thank you very much, I got it

I did it like this: AC(11|4|5) = (cx-(-2)|cy-2|cz-3), then separated it in 3 simple equations: cx+2=11; cy-2=4; cy-3=5. And got C(9|6|8).

Also, you kind of implied with your last post, that it could be measured with help of the vector's lenght, how?
 
Last edited:
  • #14
H4NS said:
Thank you very much, I got it

I did it like this: AC(11|4|5) = (cx-(-2)|cy-2|cz-3), then separated it in 3 simple equations: cx+2=11; cy-2=4; cy-3=5. And got C(9|6|8).

Also, you kind of implied with your last post, that it could be measured with help of the vector's lenght, how?

When you added the two vectors from the sides to get a vector in the direction of the diagonal, you needed to check its length to make sure it is the right length for the diagonal. In this case it checked OK. Had it not been the right length, when you added its components to the coordinates of A, you wouldn't have gotten the coordinates of C.
 
  • #15
H4NS said:
Thank you very much, I got it

I did it like this: AC(11|4|5) = (cx-(-2)|cy-2|cz-3), then separated it in 3 simple equations: cx+2=11; cy-2=4; cy-3=5. And got C(9|6|8).

Also, you kind of implied with your last post, that it could be measured with help of the vector's lenght, how?

You are making things much more complicated than they need to be. Just look at the displacement from A to B; that is, starting from A, how do you reach B? You start from x = -2 at A and get to x = +2 at B, so you go 4 units in the x-direction. You start from y = 2 at A and get to y = 10 at B, so go 8 units in the y-direction. You start at z = 3 and end at z = 4, so go 1 unit in the z-direction. So, AB = (4,8,1). Thus, in order that ABCD be a parallelogram you need DC = AB --- draw a picture to see this --- so DC (4,8,1), hence C = D + DC = (5,-2,7) + (4,8,1) = (9,6,8).

RGV
 

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