Difference between revisions of "2001 AIME II Problems/Problem 6"
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== Problem == | == Problem == | ||
− | Square <math>ABCD</math> is inscribed in a circle. Square <math>EFGH</math> has vertices <math>E</math> and <math>F</math> on <math>\overline{CD}</math> and vertices <math>G</math> and <math>H</math> on the circle. The ratio of the area of square <math>EFGH</math> to the area of square <math>ABCD</math> can be expressed as <math>\frac {m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers and <math>m < n</math>. Find <math>10n + m</math>. | + | [[Square]] <math>ABCD</math> is inscribed in a [[circle]]. Square <math>EFGH</math> has vertices <math>E</math> and <math>F</math> on <math>\overline{CD}</math> and vertices <math>G</math> and <math>H</math> on the circle. The [[ratio]] of the area of square <math>EFGH</math> to the area of square <math>ABCD</math> can be expressed as <math>\frac {m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers and <math>m < n</math>. Find <math>10n + m</math>. |
== Solution == | == Solution == | ||
− | {{ | + | Let <math>O</math> be the center of the circle, and <math>2a</math> be the side length of <math>ABCD</math>, <math>2b</math> be the side length of <math>EFGH</math>. By the [[Pythagorean Theorem]], the radius of <math>\odot O = OC = a\sqrt{2}</math>. |
+ | |||
+ | <center><asy> | ||
+ | size(150); pointpen = black; pathpen = black+linewidth(0.7); pen d = linetype("4 4") + blue + linewidth(0.7); | ||
+ | pair C=(1,1), D=(1,-1), B=(-1,1), A=(-1,-1), E= (1, -0.2), F=(1, 0.2), G=(1.4, 0.2), H=(1.4, -0.2); | ||
+ | D(MP("A",A)--MP("B",B,N)--MP("C",C,N)--MP("D",D)--cycle); D(MP("E",E,SW)--MP("F",F,NW)--MP("G",G,NE)--MP("H",H,SE)--cycle); D(CP(D(MP("O",(0,0))), A)); | ||
+ | |||
+ | D((0,0) -- (2^.5, 0), d); D((0,0) -- G -- (G.x,0), d); | ||
+ | </asy></center> | ||
+ | |||
+ | Now consider [[right triangle]] <math>OGI</math>, where <math>I</math> is the midpoint of <math>\overline{GH}</math>. Then, by the Pythagorean Theorem, | ||
+ | |||
+ | <cmath>\begin{align*} | ||
+ | OG^2 = 2a^2 &= OI^2 + GI^2 = (a+2b)^2 + b^2 \ | ||
+ | 0 &= a^2 - 4ab - 5b^2 = (a - 5b)(a + b) | ||
+ | \end{align*} | ||
+ | </cmath> | ||
+ | |||
+ | Thus <math>a = 5b</math> (since lengths are positive, we discard the other root). The ratio of the areas of two similar figures is the square of the ratio of their corresponding side lengths, so <math>\frac{[EFGH]}{[ABCD]} = \left(\frac 15\right)^2 = \frac{1}{25}</math>, and the answer is <math>10n + m = \boxed{251}</math>. | ||
== See also == | == See also == | ||
{{AIME box|year=2001|n=II|num-b=5|num-a=7}} | {{AIME box|year=2001|n=II|num-b=5|num-a=7}} | ||
+ | |||
+ | [[Category:Intermediate Geometry Problems]] |
Revision as of 12:47, 26 July 2008
Problem
Square is inscribed in a circle. Square
has vertices
and
on
and vertices
and
on the circle. The ratio of the area of square
to the area of square
can be expressed as
where
and
are relatively prime positive integers and
. Find
.
Solution
Let be the center of the circle, and
be the side length of
,
be the side length of
. By the Pythagorean Theorem, the radius of
.
![[asy] size(150); pointpen = black; pathpen = black+linewidth(0.7); pen d = linetype("4 4") + blue + linewidth(0.7); pair C=(1,1), D=(1,-1), B=(-1,1), A=(-1,-1), E= (1, -0.2), F=(1, 0.2), G=(1.4, 0.2), H=(1.4, -0.2); D(MP("A",A)--MP("B",B,N)--MP("C",C,N)--MP("D",D)--cycle); D(MP("E",E,SW)--MP("F",F,NW)--MP("G",G,NE)--MP("H",H,SE)--cycle); D(CP(D(MP("O",(0,0))), A)); D((0,0) -- (2^.5, 0), d); D((0,0) -- G -- (G.x,0), d); [/asy]](http://latex.artofproblemsolving.com/a/1/4/a14d609f4250e7310e06648a5d575443252da2bb.png)
Now consider right triangle , where
is the midpoint of
. Then, by the Pythagorean Theorem,
Thus (since lengths are positive, we discard the other root). The ratio of the areas of two similar figures is the square of the ratio of their corresponding side lengths, so
, and the answer is
.
See also
2001 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |