Difference between revisions of "2015 AMC 8 Problems/Problem 25"
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===Solution 1=== | ===Solution 1=== | ||
| − | + | We draw a diagram as shown. | |
<asy> | <asy> | ||
draw((0,0)--(0,5)--(5,5)--(5,0)--cycle); | draw((0,0)--(0,5)--(5,5)--(5,0)--cycle); | ||
Revision as of 15:51, 25 November 2015
One-inch squares are cut from the corners of this 5 inch square. What is the area in square inches of the largest square that can be fitted into the remaining space?
![[asy] draw((0,0)--(0,5)--(5,5)--(5,0)--cycle); filldraw((0,4)--(1,4)--(1,5)--(0,5)--cycle, gray); filldraw((0,0)--(1,0)--(1,1)--(0,1)--cycle, gray); filldraw((4,0)--(4,1)--(5,1)--(5,0)--cycle, gray); filldraw((4,4)--(4,5)--(5,5)--(5,4)--cycle, gray); [/asy]](http://latex.artofproblemsolving.com/9/f/2/9f2a6546513bf520224bbd20f8b0204324baaf6e.png)
Solution 1
We draw a diagram as shown.
![[asy] draw((0,0)--(0,5)--(5,5)--(5,0)--cycle); filldraw((0,4)--(1,4)--(1,5)--(0,5)--cycle, gray); filldraw((0,0)--(1,0)--(1,1)--(0,1)--cycle, gray); filldraw((4,0)--(4,1)--(5,1)--(5,0)--cycle, gray); filldraw((4,4)--(4,5)--(5,5)--(5,4)--cycle, gray); path arc = arc((2.5,4),1.5,0,90); pair P = intersectionpoint(arc,(0,5)--(5,5)); pair Pp=rotate(90,(2.5,2.5))*P, Ppp = rotate(90,(2.5,2.5))*Pp, Pppp=rotate(90,(2.5,2.5))*Ppp; draw(P--Pp--Ppp--Pppp--cycle); [/asy]](http://latex.artofproblemsolving.com/3/e/8/3e874a5e571ccf35b0e76aec24d330f7b5df7a59.png)
Let us focus on the big triangles taking up the rest of the space. The triangles on top of the unit square between the inscribed square, are similiar to the 4 big triangles by AA. Let the height of a big triangle be 
then 
.
![\[x=-x^2+6x-5\]](http://latex.artofproblemsolving.com/c/7/7/c77d96e62cdcd23bf81e073e0148887e1bf01d80.png)
![\[x^2-5x+5=0\]](http://latex.artofproblemsolving.com/0/f/2/0f26da441af2f91c38eb4fb47b5886d5da9775b0.png)
![\[x=\dfrac{5\pm \sqrt{(-5)^2-(4)(1)(5)}}{2}\]](http://latex.artofproblemsolving.com/9/4/6/946c0be301e9cbfa5e5802f3eaa1f29d4083b560.png)
![\[x=\dfrac{5\pm \sqrt{5}}{2}\]](http://latex.artofproblemsolving.com/6/1/3/61327043a82f4aa7c550aee56bbea0ba47e9a91e.png)
Which means 
This means the area of each triangle is 
This the area of the square is 
Solution 2
We draw a diagram as shown:
![[asy] pair Q,R,S,T; Q=(1.381966,0); R=(5,1.381966); S=(3.618034,5); T=(0,3.618034); draw((0,0)--(0,5)--(5,5)--(5,0)--cycle); filldraw((0,4)--(1,4)--(1,5)--(0,5)--cycle, gray); filldraw((0,0)--(1,0)--(1,1)--(0,1)--cycle, gray); filldraw((4,0)--(4,1)--(5,1)--(5,0)--cycle, gray); filldraw((4,4)--(4,5)--(5,5)--(5,4)--cycle, gray); draw(Q--R--S--T--cycle); draw((1,1)--(4,1)--(4,4)--(1,4)--cycle,dashed); [/asy]](http://latex.artofproblemsolving.com/d/7/6/d76c685d017f0721295a7bed82d367e053d88159.png)
We wish to find the area of the larger triangle. The area of the larger square is composed of the smaller square and the four triangles. The triangles have base 
and height 
, so the combined area of the four triangles is 
. The area of the smaller square is 
. We add these to see that the area of the large square is 
.
