MathDB

8

Part of 2005 AMC 10

Problems(2)

Square and inner square

Source:

11/7/2005
Square EFGH EFGH is inside the square ABCD ABCD so that each side of EFGH EFGH can be extended to pass through a vertex of ABCD ABCD. Square ABCD ABCD has side length 50 \sqrt {50} and BE \equal{} 1. What is the area of the inner square EFGH EFGH? [asy]unitsize(4cm); defaultpen(linewidth(.8pt)+fontsize(10pt));
pair D=(0,0), C=(1,0), B=(1,1), A=(0,1); pair F=intersectionpoints(Circle(D,2/sqrt(5)),Circle(A,1))[0]; pair G=foot(A,D,F), H=foot(B,A,G), E=foot(C,B,H);
draw(A--B--C--D--cycle); draw(D--F); draw(C--E); draw(B--H); draw(A--G);
label("AA",A,NW); label("BB",B,NE); label("CC",C,SE); label("DD",D,SW); label("EE",E,NNW); label("FF",F,ENE); label("GG",G,SSE); label("HH",H,WSW);[/asy]<spanclass=′latex−bold′>(A)</span> 25<spanclass=′latex−bold′>(B)</span> 32<spanclass=′latex−bold′>(C)</span> 36<spanclass=′latex−bold′>(D)</span> 40<spanclass=′latex−bold′>(E)</span> 42 <span class='latex-bold'>(A)</span>\ 25\qquad <span class='latex-bold'>(B)</span>\ 32\qquad <span class='latex-bold'>(C)</span>\ 36\qquad <span class='latex-bold'>(D)</span>\ 40\qquad <span class='latex-bold'>(E)</span>\ 42
geometrysymmetryPythagorean Theorem
Floor Tile Pattern

Source:

1/15/2009
An 8 8-foot by 10 10-foot floor is tiled with square tiles of size 1 1 foot by 1 1 foot. Each tile has a pattern consisting of four white quarter circles of radius 1/2 1/2 foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded? [asy]unitsize(2cm); defaultpen(linewidth(.8pt));
fill(unitsquare,gray); filldraw(Arc((0,0),.5,0,90)--(0,0)--cycle,white,black); filldraw(Arc((1,0),.5,90,180)--(1,0)--cycle,white,black); filldraw(Arc((1,1),.5,180,270)--(1,1)--cycle,white,black); filldraw(Arc((0,1),.5,270,360)--(0,1)--cycle,white,black);[/asy] (A)\ 80\minus{}20\pi \qquad (B)\ 60\minus{}10\pi \qquad (C)\ 80\minus{}10\pi \qquad (D)\ 60\plus{}10\pi \qquad (E)\ 80\plus{}10\pi
geometry