Difference between revisions of "2014 AIME I Problems/Problem 13"
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Let <math>s</math> be the side length of <math>ABCD</math>, let <math>Q</math>, and <math>R</math> be the midpoints of <math>\overline{EG}</math> and <math>\overline{FH}</math>, respectively, let <math>S</math> be the foot of the perpendicular from <math>Q</math> to <math>\overline{CD}</math>, let <math>T</math> be the foot of the perpendicular from <math>R</math> to <math>\overline{AD}</math>. | Let <math>s</math> be the side length of <math>ABCD</math>, let <math>Q</math>, and <math>R</math> be the midpoints of <math>\overline{EG}</math> and <math>\overline{FH}</math>, respectively, let <math>S</math> be the foot of the perpendicular from <math>Q</math> to <math>\overline{CD}</math>, let <math>T</math> be the foot of the perpendicular from <math>R</math> to <math>\overline{AD}</math>. | ||
− | <asy> size(150); defaultpen(fontsize(10pt)); pair A,B,C,D,E,F,Fp,G,Gp,H,O,I,J,R,S,T; A=dir(45*3); B=dir(-45*3); C=dir(-45); D=dir(45); O = origin; real theta=15; E=extension(A,B,O,dir(180+theta)); G=extension(C,D,O,dir(theta)); I=extension(A,D,O,dir(90+theta)); J=extension(B,C,O,dir(-90+theta)); H=(A+I)/2; F=H+(J-I); R=midpoint(H--F); S=midpoint(C--D); T=(R.x, A.y); draw(A--B--C--D--cycle^^E--G^^F--H, black+0.8); draw(S--R--T, gray+0.4); dotfactor = 3; dot("$A$",A,dir(135)); dot("$B$",B,dir(215)); dot("$C$",C,dir(305)); dot("$D$",D,dir(45)); dot("$H$",H,dir(90)); dot("$F$",F,dir(270)); dot("$E$",E,dir(180)); dot("$G$",G,dir(0)); dot("$Q$",O,dir( | + | <asy> size(150); defaultpen(fontsize(10pt)); pair A,B,C,D,E,F,Fp,G,Gp,H,O,I,J,R,S,T; A=dir(45*3); B=dir(-45*3); C=dir(-45); D=dir(45); O = origin; real theta=15; E=extension(A,B,O,dir(180+theta)); G=extension(C,D,O,dir(theta)); I=extension(A,D,O,dir(90+theta)); J=extension(B,C,O,dir(-90+theta)); H=(A+I)/2; F=H+(J-I); R=midpoint(H--F); S=midpoint(C--D); T=(R.x, A.y); draw(A--B--C--D--cycle^^E--G^^F--H, black+0.8); draw(S--R--T, gray+0.4); dotfactor = 3; dot("$A$",A,dir(135)); dot("$B$",B,dir(215)); dot("$C$",C,dir(305)); dot("$D$",D,dir(45)); dot("$H$",H,dir(90)); dot("$F$",F,dir(270)); dot("$E$",E,dir(180)); dot("$G$",G,dir(0)); dot("$Q$",O,dir(-90)); dot("$R$",R,dir(-180)); dot("$S$",S,dir(0)); dot("$T$",T,dir(90)); pair P = extension(F,H,E,G); dot("$P$",P,dir(180+60)); </asy> |
The fraction of the area of the square <math>ABCD</math> which is occupied by trapezoid <math>BCGE</math> is <cmath>\frac{275+405}{269+275+405+411}=\frac 12,</cmath>so <math>Q</math> is the center of <math>ABCD</math>. Thus <math>R</math>, <math>Q</math>, <math>S</math> are collinear, and <math>RT=QS=\tfrac 12 s</math>. Similarly, the fraction of the area occupied by trapezoid <math>CDHF</math> is <math>\tfrac 35</math>, so <math>RS=\tfrac 35s</math> and <math>RQ=\tfrac{1}{10}s</math>. | The fraction of the area of the square <math>ABCD</math> which is occupied by trapezoid <math>BCGE</math> is <cmath>\frac{275+405}{269+275+405+411}=\frac 12,</cmath>so <math>Q</math> is the center of <math>ABCD</math>. Thus <math>R</math>, <math>Q</math>, <math>S</math> are collinear, and <math>RT=QS=\tfrac 12 s</math>. Similarly, the fraction of the area occupied by trapezoid <math>CDHF</math> is <math>\tfrac 35</math>, so <math>RS=\tfrac 35s</math> and <math>RQ=\tfrac{1}{10}s</math>. | ||
Revision as of 19:44, 5 January 2022
Contents
Problem 13
On square , points
, and
lie on sides
and
respectively, so that
and
. Segments
and
intersect at a point
, and the areas of the quadrilaterals
and
are in the ratio
Find the area of square
.
Solution (Official Solution, MAA)
Let be the side length of
, let
, and
be the midpoints of
and
, respectively, let
be the foot of the perpendicular from
to
, let
be the foot of the perpendicular from
to
.
The fraction of the area of the square
which is occupied by trapezoid
is
so
is the center of
. Thus
,
,
are collinear, and
. Similarly, the fraction of the area occupied by trapezoid
is
, so
and
.
Because , the area of
is the sum
Rectangle
has area
. If
, then
has area
Therefore
.
Because these areas are in the ratio , it follows that
from which we get
. Note that
, so
and
. Then
Solution 1
Notice that . This means
passes through the center of the square.
Draw with
on
,
on
such that
and
intersects at the center of the square which I'll label as
.
Let the area of the square be
. Then
and
. This is because
is perpendicular to
(given in the problem), so
is also perpendicular to
. These two orthogonal lines also pass through the center of the square, so they split it into 4 congruent quadrilaterals.
Let the side length of the square be
.
Draw and intersects
at
. Then
Then
, so
.
Let
. Then
Consider the area of .
Thus, . Now we solve
to get
or
.
The former leads to a square with diagonal less than , which can't be, since
; therefore
and the area of
Solution 2 (Lazy)
, a multiple of
. In addition,
, which is
.
Therefore, we suspect the square of the "hypotenuse" of a right triangle, corresponding to
and
must be a multiple of
. All of these triples are primitive:
The sides of the square can only equal the longer leg, or else the lines would have to extend outside of the square. Substituting :
Thus, is the only valid answer.
Solution 3
Continue in the same way as solution 1 to get that has area
, and
. You can then find
has length
.
Then, if we drop a perpendicular from to
at
, We get
.
Thus, , and we know
, and
. Thus, we can set up an equation in terms of
using the Pythagorean theorem.
is extraneous, so
. Since the area is
, we have it is equal to
-Alexlikemath
See also
2014 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.