Difference between revisions of "User talk:Bobthesmartypants/Sandbox"
(→circles) |
(→circles) |
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Line 413: | Line 413: | ||
draw((-2,1.4)--(-1,1.4)); | draw((-2,1.4)--(-1,1.4)); | ||
label("1", (-1.5,1.4),dir(90)); | label("1", (-1.5,1.4),dir(90)); | ||
+ | </asy> | ||
+ | |||
+ | ==more circles== | ||
+ | |||
+ | <asy> | ||
+ | draw(Circle((0,0),20)); | ||
+ | draw(Circle((0,0),14)); | ||
+ | dot((0,0)); | ||
+ | dot(20*dir(60)); | ||
+ | dot(14*dir(180)); | ||
+ | dot((-17,29.445)); | ||
+ | draw(20*dir(60)--14*dir(180)--(-17,29.445)--cycle); | ||
+ | label("O",(0,0),dir(0)); | ||
+ | label("A",20*dir(60),dir(60)); | ||
+ | label("B",(-14,0),dir(180)); | ||
+ | label("P",(-17,29.445),dir(180)); | ||
+ | draw((-17,29.445)--(0,0),red); | ||
</asy> | </asy> |
Revision as of 21:43, 1 June 2014
Contents
Bobthesmartypants's Sandbox
Solution 1
First, continue to hit
at
. Also continue
to hit
at
.
We have that . Because
, we have
.
Similarly, because , we have
.
Therefore, .
We also have that because
is a parallelogram, and
.
Therefore, . This means that
, so
.
Therefore, .
Solution 2
Note that is rational and
is not divisible by
nor
because
.
This means the decimal representation of is a repeating decimal.
Let us set as the block that repeats in the repeating decimal:
.
( written without the overline used to signify one number so won't confuse with notation for repeating decimal)
The fractional representation of this repeating decimal would be .
Taking the reciprocal of both sides you get .
Multiplying both sides by gives
.
Since we divide
on both sides of the equation to get
.
Because is not divisible by
(therefore
) since
and
is prime, it follows that
.
Picture 1
Picture 2
physics problem
Solution
inscribed triangle
moar images
yay
solution reflection
origami
combos
circles
more circles