Difference between revisions of "User talk:Bobthesmartypants/Sandbox"
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==sandbox== | ==sandbox== | ||
<asy> | <asy> | ||
− | pair H,S,X; | + | unitsize(0.2mm); |
+ | pair H,S,X,A,B; | ||
H = (25,0); | H = (25,0); | ||
S = (0,115); | S = (0,115); | ||
− | + | X = (122,26); | |
+ | A = ((H+X)/2); | ||
+ | B = ((S+X)/2); | ||
draw(Circle((25,0),100)); | draw(Circle((25,0),100)); | ||
draw(Circle((0,115),150)); | draw(Circle((0,115),150)); | ||
− | draw(H--S--X--cycle, | + | draw(H--S--X--cycle); |
+ | label("100",A,dir(-90)); | ||
+ | label("150",B,dir(-120)); | ||
+ | label("H",H,dir(180)); | ||
+ | label("S",S,dir(90)); | ||
+ | label("X",X,dir(0)); | ||
</asy> | </asy> |
Revision as of 17:10, 8 December 2013
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
sandbox