Question

Geometry Proof (for fun)

Answer 1

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

Answer 2

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

Answer 3

Good answer

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

Answer 4

look forward to a cool solution :)

Answer 5

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

Answer 6

we can draw another triangle like so:

Answer 7

now if we shade in these two regions:

Answer 8

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

Answer 9

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

Answer 10

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

Answer 11

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

Answer 12

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

Answer 13

V

that is super-smart !!

Answer 14

classy :)