Find smallest area of rectangle based on lines inside.

[MeatySteak]

Hi,
I'm trying to figure out which values make up the least amount of area within the rectangle based off the lines within the rectangle? (Hopefully that makes sense)

The lines inside rotate around point E and dictate the width and length of the rectangle at the ends. Each picture is the sequence of the T shape rotating around E within the rectangle. Hopefully this makes sense.

 

[The Highlander]

Hi,
I'm trying to figure out which values make up the least amount of area within the rectangle based off the lines within the rectangle? (Hopefully that makes sense)

The lines inside rotate around point E and dictate the width and length of the rectangle at the ends. Each picture is the sequence of the T shape rotating around E within the rectangle. Hopefully this makes sense.

View attachment 37033View attachment 37034View attachment 37035

I am not at all sure that I do understand your description of what is happening here but, as far as I can figure it out, it seems to me that the area of your rectangle might be evaluated by:-

[math]65cos\theta\times 83cos\theta=5935cos^2\theta~:0°\leq\theta\leq45°[/math]
As the angle of rotation varies (from zero) up to 45°, the area of the rectangle reduces after which it increases again until it gets back to its maximum area (65 × 83 = 5935 sq. units) each time it has completed a full quarter turn, ie: rotated through 90°.

However, since \(\displaystyle cos45°\) \(\displaystyle =\frac{1}{\sqrt{2}}\), then your minimum value would just be half the maximum, ie: 2967.5 sq. units.

Perhaps some others can offer a different interpretation?

Hope that helps.

 

[The Highlander]

Hmm.

There appears to be an inherent contradiction in your description of what is happening; at least in my interpretation of it.

When I try to replicate your rotation of the "T", I am unable to get your middle drawing; the best I can do is to surround the "T" by a non-rectangular parallelogram. (See below.)

@MeatySteak. Can you explain in any more detail exactly what is going on here?

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Assuming a rotation of 30°, your 2nd diagram gives two different values for the length of the green dotted line I've added so something is definitely amiss here.

​

 

[Steven G]

Hi,
I'm trying to figure out which values make up the least amount of area within the rectangle based off the lines within the rectangle? (Hopefully that makes sense)

The lines inside rotate around point E and dictate the width and length of the rectangle at the ends. Each picture is the sequence of the T shape rotating around E within the rectangle. Hopefully this makes sense.

View attachment 37033View attachment 37034

If you rotate about point e, then point e should NOT move. So please try explaining again.

 

[The Highlander]

If you rotate about point e, then point e should NOT move. So please try explaining again.

That was the first thing that confounded me but I then assumed that s/he meant that the "T" had fixed dimensions (30+35 & 83) and, as it rotated about E, the rectangle was to be 'circumscribed' around its end points.

However, it was only after I'd tried to answer the question (derive its minimum area) that it dawned on me there was something not right about that idea, hence my attempt to replicate the 2nd figure and then proving that, with the given dimensions, it was a physical impossibility!

I suspect s/he has just drawn the second rectangle and fitted a "T" into it without checking that said "T" would remain the same size (as shown).

S/he definitely needs to explain it better unless the concept is just a figment of his/her imagination.

 

[The Highlander]

Correction (to what I said in Post #5, above):-

The second figure shown in the OP is not a physical impossibility but it is only possible if the rotation about E is c.22.8645°. It is a physical impossibility for any other rotation about E, eg: 30°, as illustrated below (and as proven by my 'green line' analysis)...

​

 

[Dr.Peterson]

I interpret the question as trying to fit a (horizontal/vertical) rectangle around the T in whatever way is possible at a given angle of rotation. Here are the rectangles for several angles:







The second original figure is just the transition between my first two.

Is this anything like the intent of the question?

Then one could find the largest rectangle of each type.

 

[The Highlander]

I interpret the question as trying to fit a (horizontal/vertical) rectangle around the T in whatever way is possible at a given angle of rotation. Here are the rectangles for several angles:

View attachment 37050

View attachment 37051

View attachment 37052

The second original figure is just the transition between my first two.

Is this anything like the intent of the question?

Then one could find the largest rectangle of each type.

Indeed, and that interpretation is implicit in the constructions I have shown at Post #6 but, if that is the case, then it is exactly why the 2nd figure in the OP is so confusing; to comply with that interpretation it should be taller and slightly narrower than the first figure (not the same height and slightly wider)!

 

[Dr.Peterson]

Indeed, and that interpretation is implicit in the constructions I have shown at Post #6 but, if that is the case, then it is exactly why the 2nd figure in the OP is so confusing; to comply with that interpretation it should be taller and slightly narrower than the first figure (not the same height and slightly wider)!

Clearly, as in many geometry problems, the pictures are not to scale. They do look as if they had been carefully made and actually measured, but that would require having solved (part of) the problem. I just don't assume drawings are to scale when I am trying to understand a problem. (I haven't put any effort yet into solving the problem, only making sure we understand it, and demonstrating all cases.)

 

[Dr.Peterson]

I put in the effort to find the dimensions of the circumscribed rectangle in each of the three cases, and determine the angles at which they transition; I won't show the details, just the results:

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As I had suspected, the answer to the initial question (smallest area of the rectangle) occurs at those transition points: exactly the three situations represented by the initial images, namely vertical, or with the end I've called C at a corner of the rectangle, or horizontal.

What surprised me was that the area in all four transition points is the same, even though the area curve is not symmetrical -- the transition angles are not complements.

Also, all these facts are the same if I change the dimensions. For example, here is the curve when 30, 35, and 83 are changed to 20, 40, and 100:

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So the more challenging question is to find the largest area of the rectangle; when I accidentally wrote "Then one could find the largest rectangle of each type," I may have subconsciously anticipated this. The smallest area is easy to calculate, though not so easy to prove!