Solving for a diagonal inside a rectangular prism?? Please help

What are your thoughts? What have you tried? How far have you gotten?

Eliz.
 
Dr.Peterson said:
I'd suggest you assign variables to the length, width, and height, and write some equations. Then write an expression for the length of AG in terms of those variables, and see what you can do. (There's a very simple answer; you don't need to solve for all the variables.)
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What i’ve done so far is:
Let width = x = AB = DC = GH = FE
Let length = y = EH = AD = BC = FG
Let height = z = AE = DH =BF = CG.
AH^2 = 6^2 = z^2 + y^2 using Pythag.
AG^2 = ?
AC^2 = x^2 + y^2
AF^2 = 4^2 = ?
And I’m stuck from here
 
greentree.23 said:
What i’ve done so far is:
Let width = x = AB = DC = GH = FE
Let length = y = EH = AD = BC = FG
Let height = z = AE = DH =BF = CG.
AH^2 = 6^2 = z^2 + y^2 using Pythag.
AG^2 = ?
AC^2 = x^2 + y^2
AF^2 = 4^2 = ?
And I’m stuck from here
Click to expand...
You should have three equations, for AC, AF, and AH. I don't know why you didn't finish AF and AC.

Then, to write an equation for AG, observe that ACG is a right triangle.

Then start putting things together.
 
Now I have AG^2 = 3^2 + z^2 with AG being the hypotenuse - I don't know what to do from here?
Dr.Peterson said:
You should have three equations, for AC, AF, and AH. I don't know why you didn't finish AF and AC.

Then, to write an equation for AG, observe that ACG is a right triangle.

Then start putting things together.
Click to expand...
 
greentree.23 said:
What i’ve done so far is:
Let width = x = AB = DC = GH = FE
Let length = y = EH = AD = BC = FG
Let height = z = AE = DH =BF = CG.
AH^2 = 6^2 = z^2 + y^2 using Pythag.
AG^2 = ?
AC^2 = x^2 + y^2
AF^2 = 4^2 = ?
And I’m stuck from here
Click to expand...
Having defined your variables, why didn't you use them?

based on this diagram...
Prism.jpg​
you could have said....

x² + y² = 3²   ⟹  \displaystyle \implies x² + y² = 9​
x² + z² = 4²   ⟹  \displaystyle \implies x² + z² = 16​
y² + z² = 6²   ⟹  \displaystyle \implies y² + z² = 36​

Let us call AG h (since it is the hypotenuse of the right-angled triangle ACG) then: h² = 3² + z²

But let us now also call AC "a" (so that a = 3 and a² = 9) but note too then that a² = x² + y²

Therefore, h² = a² + z²   ⟹  \displaystyle \implies h² = x² + y² + z²

It is possible to manipulate the three equations above to arrive at an algebraic answer for h (Hint: start by adding them together) but it should be noted, however, that using the equations to actually solve for x, y & z leads to a contradiction which means that this prism is a physical impossibility!

Notwithstanding that fact, all you are asked to do is arrive at a "value" for h (AG) so I suppose you can just proceed as suggested to arrive at one. ?
 
This doesn't seem to be possible.
I will not show my work as I want the OP to solve this problem.
 
Steven G said:
This doesn't seem to be possible.
I will not show my work as I want the OP to solve this problem.
Click to expand...
I'm not too sure on how to manipulate the other equations to solve side AG using Pythagoras. 3^2 + z^2 = AG^2 and I'm stuck from here?
 
Steven G said:
This doesn't seem to be possible.
I will not show my work as I want the OP to solve this problem.
Click to expand...
It is possible to solve it algebraically (body diagonal of a cuboid) but the prism just can't exist with those dimensions.
 
The Highlander said:
It is possible to solve it algebraically (body diagonal of a cuboid) but the prism just can't exist with those dimensions.
Click to expand...
You lost me. If the prism can't exist, then how can you find the diagonal of the prism, you know, the one that doesn't exist??
 
Steven G said:
You lost me. If the prism can't exist, then how can you find the diagonal of the prism, you know, the one that doesn't exist??
Click to expand...
I can't tell you that for four days or stapel will delete me! ?

However, (as suggested above) add the equations & simplify. ?

(I presume that (like me) you found x2 to be negative hence your comment: "This doesn't seem to be possible." but using the formula for the body diagonal of a cuboid* does produce a value for the length of AG despite the impossibility of a prism with the given dimensions. ?‍♂️)

* See Post #6.
 
I agree with @The Highlander:
2x2=−11=(x2+y2)+(x2+y2)−(y2+z2)=9+16−362x^2 = -11 = (x^2+y^2) + (x^2 + y^2) - (y^2+z^2) = 9+16-36
 
The Highlander said:
I can't tell you that for four days or stapel will delete me! ?

but using the formula for the body diagonal of a cuboid* does produce a value for the length of AG despite the impossibility of a prism with the given dimensions. ?‍♂️)
Click to expand...
I didn't know that stapel had the ability to delete people. I'm glad that I've been nice to her.

How can you find the length of AG, A and G being points in a rectangular prism, if the rectangular prism doesn't exist?

I'm not saying that you're wrong but am saying that I don't understand how you can be correct.

Sometimes I am completely confused with math and other times I just simply love math.
 
Steven G said:
How can you find the length of AG, A and G being points in a rectangular prism, if the rectangular prism doesn't exist?

I'm not saying that you're wrong but am saying that I don't understand how you can be correct.

Sometimes I am completely confused with math and other times I just simply love math.
Click to expand...
I have long found it interesting that there are some problems like this for which there is a shortcut to the solution that bypasses work that shows whether there is in fact a solution at all, and just finds the solution -- even when it does not really exist. Unfortunately, I have never made a list of such problems, so I can never point to one.

In this case, we can just add the three equations x2+y2=9x2+z2=16y2+z2=36x^2+y^2=9\\x^2+z^2=16\\y^2+z^2=36 together, and divide by 2, to get x2+y2+z2=30.5x^2+y^2+z^2=30.5 which provides an answer, but since we never actually solved for the individual variables, we don't realize (at least, I did not) that all we've proved is that if there is a solution, that would be it.

Sometimes existence has to be proved separately!
 
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