Difference between revisions of "2003 AMC 12A Problems/Problem 14"
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First we show that <math>KLMN</math> is a square. How? Show that it is a rhombus that has a right angle. | First we show that <math>KLMN</math> is a square. How? Show that it is a rhombus that has a right angle. | ||
| − | <math>\angle{NAK}=\angle{KBL}=\angle{LCM}=\angle{DNM}=150^\circ</math>. All the sides of the equilateral triangles are equal, so the triangles are congruent. Notice that <math>\angle{KNL}=45^\circ</math>, etc, so <math>\angle{KNM}=90^circ</math>. So we have a square. | + | <math>\angle{NAK}=\angle{KBL}=\angle{LCM}=\angle{DNM}=150^\circ</math>. All the sides of the equilateral triangles are equal, so the triangles are congruent. Notice that <math>\angle{KNL}=45^\circ</math>, etc, so <math>\angle{KNM}=90^\circ</math>. So we have a square. |
Instead of going to find <math>KN</math> using Law of Cosines, we can inscribe the square in another bigger one and then subtract the four right triangles in the corners, so after all of this, we find that the answer is <math>\boxed{\textbf{(D) } 32+16\sqrt{3}}</math> | Instead of going to find <math>KN</math> using Law of Cosines, we can inscribe the square in another bigger one and then subtract the four right triangles in the corners, so after all of this, we find that the answer is <math>\boxed{\textbf{(D) } 32+16\sqrt{3}}</math> | ||
Latest revision as of 16:26, 22 June 2023
Problem
Points 
and 
lie in the plane of the square 
such that 
, 
, 
, and 
are equilateral triangles. If has an area of 16, find the area of 
.
![[asy] unitsize(2cm); defaultpen(fontsize(8)+linewidth(0.8)); pair A=(-0.5,0.5), B=(0.5,0.5), C=(0.5,-0.5), D=(-0.5,-0.5); pair K=(0,1.366), L=(1.366,0), M=(0,-1.366), N=(-1.366,0); draw(A--N--K--A--B--K--L--B--C--L--M--C--D--M--N--D--A); label("$A$",A,SE); label("$B$",B,SW); label("$C$",C,NW); label("$D$",D,NE); label("$K$",K,NNW); label("$L$",L,E); label("$M$",M,S); label("$N$",N,W); [/asy]](http://latex.artofproblemsolving.com/1/4/0/14057a8f4f08bad17471fdaed56a758f1e2fcea7.png)
Solution
Solution 1
Since the area of square 
is 16, the side length must be 4. Thus, the side length of triangle AKB is 4, and the height of 
, and thus 
, is 
.
The diagonal of the square 
will then be 
. From here there are 2 ways to proceed:
First: Since the diagonal is , the side length is 
, and the area is thus 
.
Second: Since a square is a rhombus, the area of the square is 
, where 
and 
are the diagonals of the rhombus. Since the diagonal is , the area is 
.
Solution 2
Because has area 
, its side length is simply 
.
Angle chasing, we find that the angle of 
.
We also know that 
.
Using Law of Cosines, we find that side 
.
However, the area of is simply 
, hence the answer is 
Solution 3
First we show that is a square. How? Show that it is a rhombus that has a right angle.
. All the sides of the equilateral triangles are equal, so the triangles are congruent. Notice that 
, etc, so 
. So we have a square.
Instead of going to find 
using Law of Cosines, we can inscribe the square in another bigger one and then subtract the four right triangles in the corners, so after all of this, we find that the answer is
~hastapasta (edited)
See Also
| 2003 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 13 |
Followed by Problem 15 |
| 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 | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
