Difference between revisions of "2015 AMC 8 Problems/Problem 25"

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This means the area of each triangle is <math>\dfrac{5-\sqrt{5}}{2}*(5-\dfrac{5-\sqrt{5}}{2})*\dfrac{1}{2}=\dfrac{5}{2}</math>
 
This means the area of each triangle is <math>\dfrac{5-\sqrt{5}}{2}*(5-\dfrac{5-\sqrt{5}}{2})*\dfrac{1}{2}=\dfrac{5}{2}</math>
 
This the area of the square is <math>25-(4*\dfrac{5}{2})=\boxed{C,~15}</math>
 
This the area of the square is <math>25-(4*\dfrac{5}{2})=\boxed{C,~15}</math>
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 +
Solution 2: 
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 +
We draw a diagram as shown:
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<asy>
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pair Q,R,S,T;
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Q=(1.381966,0);
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R=(5,1.381966);
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S=(3.618034,5);
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T=(0,3.618034);
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draw((0,0)--(0,5)--(5,5)--(5,0)--cycle);
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filldraw((0,4)--(1,4)--(1,5)--(0,5)--cycle, gray);
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filldraw((0,0)--(1,0)--(1,1)--(0,1)--cycle, gray);
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filldraw((4,0)--(4,1)--(5,1)--(5,0)--cycle, gray);
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filldraw((4,4)--(4,5)--(5,5)--(5,4)--cycle, gray);
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draw(Q--R--S--T--cycle);
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draw((1,1)--(4,1)--(4,4)--(1,4)--cycle,dashed);
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</asy>
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 +
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 <math>3</math> and height <math>1</math>, so the combined area of the four triangles is <math>4 \cdot \frac 32=6</math>.  The area of the smaller square is <math>9</math>.  We add these to see that the area of the large square is <math>9+6=\boxed{\mathrm{(C) \ } 15}</math>.

Revision as of 16:48, 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?

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

[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

Lets draw a diagram. [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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