Pictures would help this thread, but I don't have paint on my computer to work with. :sarcastic:
I was introduced today to special right triangles- 30.60.90 and 45.45.90
A while ago I gave a triangle(?) sides dealing with infinity.
The triangle I created is a right triangle. One leg of the triangle is ∞ in length. The other leg's length is 1/∞ (or 0.∞). With a simple Pythagorean theorem I found the hypotenuse to be ∞.∞ in length. I wasn't thinking about the angles before, but with one leg of the triangle being so infinitely small, the angle must be very small as well, small but not infinitely small because the triangle is right and the angles beside the right angle must add up to 90. The angle opposite the infinitely small side is, I guess, 0.(lots of 0s)1 and the angle opposite the ∞ leg must be 89.9 repeating.
Anyways, today I learned about these special right triangles and will now add infinity to them and their specialness.
First I'll play with the 30.60.90 triangle.
Quick calculations messes with the triangle. The hypotenuse is 2*∞, which is ∞ [when starting out with ∞ as the short leg***]. The long leg is ∞*the square root of 3. Might be my faulty workings with ∞; I'll come back to this one later.
Now for the 45.45.90, isosceles!
The converse of the Isosceles Triangles Base Angles Theorem states that
If two angles of a triangle are congruent, then sides opposite those angles are congruent. (1)
For my experiment the legs will be ∞ in length. The hypotenuse of this 45.45.90 must be ∞*the square root of 2.
Can be about my use of infinity, infinity in general (this is only partly an infinity thread), multiplication/addition/subtraction/division and infinity, and anything else infinite. We can also discuss the triangles I've presented, or, if you have the time and patience, other shapes and infinity.
I'd like to read your thoughts. Recreational math is fun.
(1) Proofs involving Isosceles triangles, theorems, examples and practice proofs
For more on special triangles:
Special Right Triangles (with worked solutions & videos)