Thread by @ProfDFrancis

Fun puzzle to limber ourselves up for tomorrow's tweetorial.

Squares have area 1 (even the purple one which is a bit long in the face).

What is the area of the big rectangle, and why?

I won't give the source yet as she has received lots of correct answers.

My way which I can understand but is not the most elegant (there are some beautiful geometric solutions).

What is the length of this hypotenuse (long side of the right angled triangle)?

Let's call that length B because it is the big bit of the long side of the rectangle.

Now we look at the small bit.

And to do this, I propose to apply Francis's first law of things where there are two different criss cross 90 degree grids at an angle from each other.

"All triangles are similar"

Euclid missed that one out of his book of good things to know, I think you'll find.

So we know the big triangle has sides in proportion (small to big) of

1
2
Sqrt 5

So has this small one.

The hypotenuse (long side) obviously has length 2.

So how long is the short side?

So that is the length of the small bit (S) of the long side of the rectangle.

The whole length of the rectangle is therefore B plus S which is:

I'm amazed everyone is getting it right.

Maybe in medicine, everyone thinks they should have an opinion, because it looks easy?

But in mathsy things, people are willing to say they haven't worked it out yet.

Ermmmm I spoke too soon.

The ratios of the sides are 1 : 2 : sqrt 5

So in the small triangle, if the long side is 2, the small side must be 2 divided by sqrt 5, yes?

And if the short side of a triangle is 2 over sqrt 5, what is the medium length side ?

Is it

Now I know some people will be tempted to pretend to forget about this little bit.

"It's pretty small, who cares, nobody will notice "

But this is maths, not medicine.

Luckily by Francis s first law of triangly grid questions, the we know ths the height of this thing must be 1/sqrt5 of the hypotenuse (which this time happens to be simply 1).

So the height of this last bit is just 1/sqrt 5.

So

Height of the rectangle =
4 / sqrt 5 plus 1 / sqrt 5
Or just 5 / sqrt 5 which is just sqrt 5, yay!

Width =
Sqrt 5 plus 2 / sqrt 5

Area is the product of these

r5 (r5 + 2/r5)
= 5 + 2
= 7

Wonderful puzzle by @Cshearer41 Catherine Shearer!

Source

https://twitter.com/Cshearer41/status/1259087361645969409?s=19

And my favourite solution?

One with no numbers at all.

THIS IS MATHEMATICS ...

Right get a good night's sleep in preparation for a hard to understand graph, on tomorrow's tweetorial.

Charlotte Manisty spent half an hour explaining it to me, but tomorrow I am aiming to make myself look as though I understood what it meant, first time.

Postscript.

I will somehow try to sleep myself, despite the horror of realising that all these people have the right to vote.