2015 AMC 8 Problems/Problem 25

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?

$\mathrm{(A) \ } 9\qquad \mathrm{(B) \ } 12\frac{1}{2}\qquad \mathrm{(C) \ } 15\qquad \mathrm{(D) \ } 15\frac{1}{2}\qquad \mathrm{(E) \ } 17$

$[asy] draw((0,0)--(0,5)--(5,5)--(5,0)--cycle); filldraw((0,4)--(1,4)--(1,5)--(0,5)--cycle, gray); filldraw((0,0)--(1,0)--(1,1)--(0,1)--cycle, gray); filldraw((4,0)--(4,1)--(5,1)--(5,0)--cycle, gray); filldraw((4,4)--(4,5)--(5,5)--(5,4)--cycle, gray); [/asy]$

Solution 1

We draw a diagram as shown. $[asy] draw((0,0)--(0,5)--(5,5)--(5,0)--cycle); filldraw((0,4)--(1,4)--(1,5)--(0,5)--cycle, gray); filldraw((0,0)--(1,0)--(1,1)--(0,1)--cycle, gray); filldraw((4,0)--(4,1)--(5,1)--(5,0)--cycle, gray); filldraw((4,4)--(4,5)--(5,5)--(5,4)--cycle, gray); path arc = arc((2.5,4),1.5,0,90); pair P = intersectionpoint(arc,(0,5)--(5,5)); pair Pp=rotate(90,(2.5,2.5))*P, Ppp = rotate(90,(2.5,2.5))*Pp, Pppp=rotate(90,(2.5,2.5))*Ppp; draw(P--Pp--Ppp--Pppp--cycle); [/asy]$ 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 $x$ then $\dfrac{x}{x-1}=\dfrac{5-x}{1}$ . $x=-x^2+6x-5$ $x^2-5x+5=0$ $x=\dfrac{5\pm \sqrt{(-5)^2-(4)(1)(5)}}{2}$ $x=\dfrac{5\pm \sqrt{5}}{2}$ Which means $x=\dfrac{5-\sqrt{5}}{2}$ This means the area of each triangle is $\dfrac{5-\sqrt{5}}{2}*(5-\dfrac{5-\sqrt{5}}{2})*\dfrac{1}{2}=\dfrac{5}{2}$ This the area of the square is $25-(4*\dfrac{5}{2})=\boxed{C,~15}$

Solution 2

We draw a diagram as shown: $[asy] pair Q,R,S,T; Q=(1.381966,0); R=(5,1.381966); S=(3.618034,5); T=(0,3.618034); draw((0,0)--(0,5)--(5,5)--(5,0)--cycle); filldraw((0,4)--(1,4)--(1,5)--(0,5)--cycle, gray); filldraw((0,0)--(1,0)--(1,1)--(0,1)--cycle, gray); filldraw((4,0)--(4,1)--(5,1)--(5,0)--cycle, gray); filldraw((4,4)--(4,5)--(5,5)--(5,4)--cycle, gray); draw(Q--R--S--T--cycle); draw((1,1)--(4,1)--(4,4)--(1,4)--cycle,dashed); [/asy]$

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 $3$ and height $1$ , so the combined area of the four triangles is $4 \cdot \frac 32=6$ . The area of the smaller square is $9$ . We add these to see that the area of the large square is $9+6=\boxed{\mathrm{(C) \ } 15}$ .

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2015 AMC 8 Problems/Problem 25

Solution 1

Solution 2