Recent content by kuruman

Here is how it can be done without Pythagoras or trigonometry. Right triangles ADC and CDB are similar. Therefore ##\dfrac{b}{h}=\dfrac{h}{a}\implies h^2=ab\implies h=\sqrt{ab}.## I am pretty sure it was proven this way in my geometry class. On edit. Segment ##h## is the geometric mean of the...

Yes, I had that one also and that is why I labeled sides ##h_1## and ##h_2##. However, I dropped it after seeing @renormalize's, more elegant proof using Pythagoras. I remember seeing this proposition in high-school geometry (before trigonometry) but I cannot remember whether it was done with...

Trigonometric proof ##\theta +\varphi=90^{\circ}## Therefore ##\tan\theta=\cot\varphi\implies \tan\theta \tan\varphi=1.## ##1=\dfrac{b}{h}\cdot \dfrac{a}{h}\implies h^2=ab\implies h=\sqrt{ab}.##

Have you not noticed how birds nonchalantly perch on high tension wires? Sure, the distance between their feet is less than the distance between the man's hands, but a bird's resistance is also much less than a man's resistance.

I understood its importance after solving it. A key is a key is a key. :oldsmile: I looked for other solutions, but I did not find any, at least with 4x4 squares.

Is this the third solution you had in mind?

Been there myself. If it helps any, I learned by experience, after losing my hard work several times, to copy and paste posts with many complex equations in a backup text file as I go along. I assemble the backup in small sections each of which I review and edit separately as if they were...

That's the first (brute force) approach I tried without much success. So I thought that there might be a clever shortcut known to experts. If there isn't, I'll try again. Thanks.

It looks like I didn't make my question clear. Here is what I meant to ask: Find integer values X ,Y, U, W and Z such that XUWZ km = YUWZ mi (rounded four digit distances) given the conversion factor 1 km = 0.621371 mi. I stumbled into one possibility. Are there more 4-digit distances...

I recently had to convert 7,923 km to miles and used the web converter (see right). It turned out that 7,923 km = 4,923 mi ignoring the decimal change. The matching of the last three digits astonished me and I wondered if there is a systematic way to find additional conversion matches like this...

This doesn't help. A cut along the red line produces two pieces that have the same shape, namely rectangles. Is this a valid solution? If not, why not?

Does "identical" mean that after the object is cut into two, one can superimpose the two pieces into one with no more than a simple rotation in the plane of the screen?

That is why I picked those particular numbers. I gave this problem to my wife and she figured it out the same way. She is a professional historian with analytical skills.

I see a lot of numbers but no answer. Will the driver be able to get out in time and why? You need to supply words explaining what the numbers mean. It looks like you used a calculator to answer this question. Here is how you can do it in your head. The train moves at 60 km/h. Since 1 hour...