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Difference between revisions of "2002 AMC 12B Problems/Problem 18"

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(→‎Solution 2)
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=== Solution 2 ===
 
=== Solution 2 ===
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 +
draw((0,0)--(6,0)--(0,3)--cycle);
 +
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
 +
label("<math>b</math>",(2.5,0),S);
 +
label("<math>a</math>",(0,1.5),W);
 +
label("<math>c</math>",(2.5,1),W);
 +
label("<math>A</math>",(0.5,2.5),W);
 +
label("<math>B</math>",(3.5,0.75),W);
 +
label("<math>C</math>",(1,1),W);
 +
</asy></center>
 +
 +
draw((0,0)--(6,0)--(6,2)--(0,2)--cycle);
 +
draw((0,2)--(0,3,5));
 +
draw((6,0)--(7.5,0));
  
 
Assume that a point <math>P</math> is randomly chosen inside the rectangle with vertices <math>(0,0)</math>, <math>(3,0)</math>, <math>(3,1)</math>, <math>(0,1)</math>.  
 
Assume that a point <math>P</math> is randomly chosen inside the rectangle with vertices <math>(0,0)</math>, <math>(3,0)</math>, <math>(3,1)</math>, <math>(0,1)</math>.  

Revision as of 15:39, 2 July 2019

Problem

A point $P$ is randomly selected from the rectangular region with vertices $(0,0),(2,0),(2,1),(0,1)$. What is the probability that $P$ is closer to the origin than it is to the point $(3,1)$?

$\mathrm{(A)}\ \frac 12 \qquad\mathrm{(B)}\ \frac 23 \qquad\mathrm{(C)}\ \frac 34 \qquad\mathrm{(D)}\ \frac 45 \qquad\mathrm{(E)}\ 1$

Solution

Solution 1

2002 12B AMC-18.png

The region containing the points closer to $(0,0)$ than to $(3,1)$ is bounded by the perpendicular bisector of the segment with endpoints $(0,0),(3,1)$. The perpendicular bisector passes through midpoint of $(0,0),(3,1)$, which is $\left(\frac 32, \frac 12\right)$, the center of the unit square with coordinates $(1,0),(2,0),(2,1),(1,1)$. Thus, it cuts the unit square into two equal halves of area $1/2$. The total area of the rectangle is $2$, so the area closer to the origin than to $(3,1)$ and in the rectangle is $2 - \frac 12 = \frac 32$. The probability is $\frac{3/2}{2} = \frac 34 \Rightarrow \mathrm{(C)}$.

Solution 2

draw((0,0)--(6,0)--(0,3)--cycle); draw((0,0)--(2,0)--(2,2)--(0,2)--cycle); label("$b$",(2.5,0),S); label("$a$",(0,1.5),W); label("$c$",(2.5,1),W); label("$A$",(0.5,2.5),W); label("$B$",(3.5,0.75),W); label("$C$",(1,1),W);

</asy>

draw((0,0)--(6,0)--(6,2)--(0,2)--cycle); draw((0,2)--(0,3,5)); draw((6,0)--(7.5,0));

Assume that a point $P$ is randomly chosen inside the rectangle with vertices $(0,0)$, $(3,0)$, $(3,1)$, $(0,1)$.

In this case, the probability that $P$ is closer to the origin than to point $(3,1)$ is $\frac{1}{2}$.

If $P$ is chosen within the square with vertices

See also

2002 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 17 		Followed by
Problem 19
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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