Difference between revisions of "User talk:Bobthesmartypants/Sandbox"
(→sandbox) |
(→sandbox) |
||
Line 78: | Line 78: | ||
<asy> | <asy> | ||
unitsize(0.2mm); | unitsize(0.2mm); | ||
− | pair H,S,X,A,B; | + | pair H,S,X,Y,A,B; |
H = (25,0); | H = (25,0); | ||
S = (0,115); | S = (0,115); | ||
X = (122,26); | X = (122,26); | ||
+ | Y = (-75,-17); | ||
A = ((H+X)/2); | A = ((H+X)/2); | ||
B = ((S+X)/2); | B = ((S+X)/2); | ||
− | draw(Circle( | + | draw(Circle(H,100)); |
− | draw(Circle( | + | draw(Circle(S,150)); |
draw(H--S--X--cycle); | draw(H--S--X--cycle); | ||
+ | draw(H--Y--S,linetype("8 8")); | ||
label("100",A,dir(-90)); | label("100",A,dir(-90)); | ||
label("150",B,dir(-120)); | label("150",B,dir(-120)); | ||
− | label("H",H,dir( | + | label("H",H,dir(-90)); |
label("S",S,dir(90)); | label("S",S,dir(90)); | ||
label("X",X,dir(0)); | label("X",X,dir(0)); | ||
+ | label("X'",Y,dir(-135)); | ||
</asy> | </asy> | ||
+ | |||
+ | '''PROVING THE EXISTENCE OF SUCH A POINT''' | ||
+ | |||
+ | We first want to prove that a point <math>X</math> exists such that <math>HX=100</math> and <math>SX=150</math>. | ||
+ | |||
+ | First we draw a circle with center <math>H</math> and radius <math>100</math>. This denotes the locus of all points <math>P</math> such that <math>HP=100</math>. | ||
+ | |||
+ | Now we draw a circle with center <math>S</math> and radius <math>150</math>. This denotes the locus of all points <math>P</math> such that <math>SP=150</math>. | ||
+ | |||
+ | Note that the intersection of these two locuses are the points which satisfy both conditions. | ||
+ | |||
+ | We see that there are two points which satisfy both locuses: <math>X</math> and <math>X'</math>. | ||
+ | |||
+ | We get rid of the extraneous solution, <math>X'</math>, because it does not satisfy the need that the treasure is on land. | ||
+ | |||
+ | Therefore the point that we seek is <math>X</math>, and we have proved its existence. <math>\Box</math> | ||
+ | |||
+ | Note that we are assuming that <math>SH\le SX+HX</math>. This is true because the diagram given is to scale. | ||
+ | |||
+ | '''PROVING THAT THE POINT IS UNIQUE''' | ||
+ | |||
+ | <asy> | ||
+ | unitsize(0.2mm); | ||
+ | pair H,S,X,Y,A,B,C; | ||
+ | H = (25,0); | ||
+ | S = (0,115); | ||
+ | X = (122,26); | ||
+ | Y = (120,50); | ||
+ | A = ((H+X)/2); | ||
+ | B = ((S+X)/2); | ||
+ | C = ((H+S)/2); | ||
+ | draw(H--S--X--cycle); | ||
+ | draw(H--Y--S,linetype("8 8")); | ||
+ | label("100",A,dir(-90)); | ||
+ | label("150",B,dir(-120)); | ||
+ | label("H",H,dir(-90)); | ||
+ | label("S",S,dir(90)); | ||
+ | label("X",X,dir(0)); | ||
+ | label("X'",Y,dir(45)); | ||
+ | label("n",C,dir(210)); | ||
+ | </asy> | ||
+ | |||
+ | Note that if there is another point <math>X'</math>, then it must satisfy that <math>HX'=100</math> and <math>SX'=150</math>. | ||
+ | |||
+ | We can let <math>SH=n</math>. The triangle <math>SHX'</math> therefore has sides of length <math>n</math>, <math>100</math>, and <math>150</math>. | ||
+ | |||
+ | However, because of SSS congruency, <math>SHX'</math> must be congruent to <math>SHX</math>. Since <math>SX'=SX</math>, then <math>X</math> and <math>X'</math> are the same point, and therefore <math>X</math> is unique. <math>\Box</math> | ||
+ | |||
+ | Note that there is an extraneous solution for <math>X'</math> that is to the left of the line <math>\overline{SH}</math>. | ||
+ | |||
+ | However, since this does not meet the requirements of the point being on land, it does not work. |
Revision as of 16:45, 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
PROVING THE EXISTENCE OF SUCH A POINT
We first want to prove that a point exists such that and .
First we draw a circle with center and radius . This denotes the locus of all points such that .
Now we draw a circle with center and radius . This denotes the locus of all points such that .
Note that the intersection of these two locuses are the points which satisfy both conditions.
We see that there are two points which satisfy both locuses: and .
We get rid of the extraneous solution, , because it does not satisfy the need that the treasure is on land.
Therefore the point that we seek is , and we have proved its existence.
Note that we are assuming that . This is true because the diagram given is to scale.
PROVING THAT THE POINT IS UNIQUE
Note that if there is another point , then it must satisfy that and .
We can let . The triangle therefore has sides of length , , and .
However, because of SSS congruency, must be congruent to . Since , then and are the same point, and therefore is unique.
Note that there is an extraneous solution for that is to the left of the line .
However, since this does not meet the requirements of the point being on land, it does not work.