Vocaloid:

Geometry Proof (for fun)

Vocaloid:

|dw:1502651945890:dw|

Vocaloid:

Assume that this rectangle is composed of 3 squares, each with equal side length. Prove that A + B + C = 90 degrees Hint: think outside the box

sillybilly123:

with $$A = {\pi \over 4}$$, need to show that $$B + C = {\pi \over 4}$$ so tan addition forms: $$\tan B = {1 \over 2}$$ $$\tan C = {1 \over 3}$$ $$\implies \tan (B +C) = \dfrac{\tan B + \tan C }{1 - \tan B \tan C} = \dfrac{\frac{1}{2} + \frac{1}{3} }{1 - \frac{1}{2} * \frac{1}{3} } = 1$$ not the beautiful solution but *a* solution

Vocaloid:

Good answer There are a lot of ways to solve this, I'll post another solution later tonight

sillybilly123:

look forward to a cool solution :)

Vocaloid:

for this solution we draw an identical rectangle above the first rectangle, partitioned in the same way

Vocaloid:

|dw:1502675191558:dw|

Vocaloid:

we can draw another triangle like so:|dw:1502675258798:dw|

Vocaloid:

now if we shade in these two regions:

Vocaloid:

|dw:1502675297868:dw|

Vocaloid:

(the graph isn't drawn well but the two black triangles are equal AND they are equal to the triangle containing angle B

Vocaloid:

|dw:1502675431468:dw|

Vocaloid:

now if we look at the big triangle in the middle, the legs are equal, making the lower right angle equal to 45

Vocaloid:

|dw:1502675506363:dw|

Vocaloid:

without knowing any trig, we can see that the original angle A is 90/2 = 45 since the right angle is bisected

Vocaloid:

|dw:1502675558025:dw|

Vocaloid:

therefore A + B + C = the angle in the lower right corner = 90 degrees

Vocaloid:

I will admit it is more work than the direct calculations but I think it is very clever @sillybilly123

sillybilly123:

V that is super-smart !!

sillybilly123:

classy :)