2002 AMC 10B Problems/Problem 17
Problem
A regular octagon 
has sides of length two. Find the area of 
.
Solution
![[asy] unitsize(1cm); defaultpen(0.8); pair[] A = new pair[8]; A[0]=(0,0); for (int i=1; i<8; ++i) A[i] = A[i-1] + 2*dir(45*(i-1)); draw( A[0]--A[1]--A[2]--A[3]--A[4]--A[5]--A[6]--A[7]--cycle ); label("$A$",A[0],SW); label("$B$",A[1],SE); label("$C$",A[2],SE); label("$D$",A[3],NE); label("$E$",A[4],NE); label("$F$",A[5],NW); label("$G$",A[6],NW); label("$H$",A[7],SW); filldraw( A[0]--A[3]--A[6]--cycle, lightgray, black ); pair P = intersectionpoint( A[3]--A[6], A[0]--A[5] ); draw( A[0]--P ); draw( P -- A[5], dashed ); label("$P$",P,NE); draw( A[1]--A[4], dashed ); pair Q = intersectionpoint( A[3]--A[6], A[1]--A[4] ); label("$Q$",Q,NW); [/asy]](http://latex.artofproblemsolving.com/c/a/9/ca90d855ac2876463b79d50db160c0d88e6f0f30.png)
The area of the triangle 
can be computed as 
. We will now find 
and 
.
Clearly, 
is a right isosceles triangle with hypotenuse of length 
, hence 
.
The same holds for triangle 
and its leg 
. The length of 
is equal to 
.
Hence 
, and 
.
Then the area of equals 
.
Solution 2 (alcmus solution)
If the mouse is at 
, then the square of the distance from the mouse to the cheese is\[
(x - 12)^2 + (8 - 5x)^2 =
26(x^2 - 4x + 8) = 26((x - 2)^2 + 4).
\]The value of this expression is smallest when 
, so the mouse is closest to the cheese at the point 
, and 
.
See Also
| 2002 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 16 |
Followed by Problem 18 | |
| 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 10 Problems and Solutions | ||
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