Construction of a rectangle

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Rebeka

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How to construct a rectangle given one of his sides(a) and the diference between the diagonal and the other side(b)?
 
Rebeka said:
How to construct a rectangle given one of his sides(a) and the diference between the diagonal and the other side(b)?
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Interesting problem! Do you have any ideas?

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I'd probably start with some algebra, trying to solve for b given a and d = c-b. Then I'd turn that algebraic solution into a construction.

Or maybe you've learned some theorem that could be of use, and we could help if you told us what you've learned lately.
 
I was thinking the same way. An idea for a more geometric solution maybe?
 
Rebeka said:
I was thinking the same way. An idea for a more geometric solution maybe?
Click to expand...
The construction will be totally geometric. Only the thinking to discover it will be algebraic.

Are you saying you want to tie one hand behind your back by forcing yourself to do all your thinking geometrically, as Euclid might? I suppose it's possible, but I have no interest in trying.
 
Dr.Peterson said:
The construction will be totally geometric. Only the thinking to discover it will be algebraic.

Are you saying you want to tie one hand behind your back by forcing yourself to do all your thinking geometrically, as Euclid might? I suppose it's possible, but I have no interest in trying.
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Thank you for your help.:) I would like to share another idea. If the rectangle is ABCD, AB=a,BC=b and BD=d, than we can prolong the side DA through the point A to a point E so AE=d-b. Now we can construct the triangle ABE given (side a,side d-b and right angle). Considering the fact that the triangle EBD is isosceles triangle( DE=d and BD=d) than the bisector of BE should pass through the point D. We get both sides of the rectangle and we can easily find the point C.
We can use the similar thinking if we have to construct a rectangle given a and d+b.
 
Rebeka said:
Thank you for your help.:) I would like to share another idea. If the rectangle is ABCD, AB=a,BC=b and BD=d, than we can prolong the side DA through the point A to a point E so AE=d-b. Now we can construct the triangle ABE given (side a,side d-b and right angle). Considering the fact that the triangle EBD is isosceles triangle( DE=d and BD=d) than the bisector of BE should pass through the point D. We get both sides of the rectangle and we can easily find the point C.
We can use the similar thinking if we have to construct a rectangle given a and d+b.
Click to expand...
Nice method; and it can indeed be discovered just by drawing the final figure and working backward, trying to get to a point that could be constructed using only what's given.

This illustrates a general observation: Many problems that can be solved by routine algebra can also be solved by non-routine thinking. One can choose between routine thinking that you know will work, and creative thinking that you hope will work (but will be more fun). In this case I was satisfied with the former because it wasn't my problem, while you were more optimistic than I was at the moment (and perhaps had the time).
 
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