Another geometry puzzle

[ceptimus]

Find the angle ϴ

triangle.png

[ceptimus]

Aren't the big one and the little one similar triangles?

I'll go with that == 20 degrees.

Spoiler:

I drew it accurately, so you can see that answer is wrong by measuring it on your screen with a protractor. The challenge, though, is to prove the value of theta using geometry or trigonometry.

It can be done with basic high school geometry - no special knowledge needed. But, of course, it's not obvious at first how to tackle it - which is what makes it an interesting puzzle.

It can also be done with high school trigonometry - but it can get a bit messy just using sines, cosines and tangents.

There's also probably a reasonably neat way to do it using basic trigonometry and simultaneous equations.

[Witness]

30°.

[Witness]

How did you get there?

https://en.wikipedia.org/wiki/Law_of_sines

[ceptimus]

Here's the way I did it, just using basic geometry.

Spoiler:

My solution on the Freethought Forum from posted 14 years ago (but I worked it out a few years before then).

Spoiler:

The puzzle is a famous one posed in the Mathematical Gazette by Edward Mann Langley in 1922

[Bruce]

Brucetta requested 1/4 slice of pizza. Bruce gives her 1/3 slice of pizza. Brucetta says that's too much. What fraction of pizza will Bruce have to remove from the 1/3 slice of pizza to leave her with 1/4?

Bonus questions! How long will it take for Bruce and Mrs. Bruce to argue over the best means solve this puzzle? How long will the argument last before Brucetta says, "Forget about it! I'll just eat the whole 1/3! "

[Witness]

Brucetta requested 1/4 slice of pizza. Bruce gives her 1/3 slice of pizza. Brucetta says that's too much. What fraction of pizza will Bruce have to remove from the 1/3 slice of pizza to leave her with 1/4?

1/12. You're joking, right?

Bonus questions! How long will it take for Bruce and Mrs. Bruce to argue over the best means solve this puzzle? How long will the argument last before Brucetta says, "Forget about it! I'll just eat the whole 1/3! "

Till late into the night, especially after lights out.

Brucetta will behave for 30 seconds (standard case) to 1 minute (on a good day).

Did I win?

[Bruce]

[gnome]

Brucetta requested 1/4 slice of pizza. Bruce gives her 1/3 slice of pizza. Brucetta says that's too much. What fraction of pizza will Bruce have to remove from the 1/3 slice of pizza to leave her with 1/4?

1/12. You're joking, right?

Trick question. Regardless of the amount chosen, it will either be too much removed or too little removed.

In fact, it can be mathematically proven in R² that the odds of slicing it at exactly the correct mark is literally zero.

[Witness]

↑ You mean like the probability of choosing any real number?

[Bruce]

Perfect excuse to keep the whole pizza to myself.

[Rob Lister]

Zeno might submit that there is no way to provide 1/4th a slice.

[ceptimus]

What's the volume of a circular pizza with radius z, and height a?

[gnome]

V=πz²a

[ceptimus]

Yes, or in plain ASCII, V = Pi z z a

[Rob Lister]

What is the most cost effective size of a pizza hut pizza.

[gnome]

Without looking, almost always the largest size.

[ceptimus]

I would say the zero-size pizza hut pizza (i.e. don't buy one from there) is the most cost-effective solution. Instead, buy one from a supermarket and warm it up yourself.

[Doctor X]

– J.D.