Difference between revisions of "1989 AHSME Problems/Problem 13"
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The rhombus has a base of length <math>\frac1{\sin\alpha}</math> and height of <math>1</math>. Its area is the product. | The rhombus has a base of length <math>\frac1{\sin\alpha}</math> and height of <math>1</math>. Its area is the product. | ||
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Revision as of 12:48, 5 July 2013
Problem
Two strips of width 1 overlap at an angle of 
as shown. The area of the overlap (shown shaded) is
![[asy] pair a = (0,0),b= (6,0),c=(0,1),d=(6,1); transform t = rotate(-45,(3,.5)); pair e = t*a,f=t*b,g=t*c,h=t*d; pair i = intersectionpoint(a--b,e--f),j=intersectionpoint(a--b,g--h),k=intersectionpoint(c--d,e--f),l=intersectionpoint(c--d,g--h); draw(a--b^^c--d^^e--f^^g--h); filldraw(i--j--l--k--cycle,blue); label("$\alpha$",i+(-.5,.2)); //commented out labeling because it doesn't look right. //path lbl1 = (a+(.5,.2))--(c+(.5,-.2)); //draw(lbl1); //label("$1$",lbl1);[/asy]](http://latex.artofproblemsolving.com/7/6/2/762566bf70020b367302b9e38d58b5e686d1c83c.png)
Solution
![[asy] pair a = (0,0),b= (6,0),c=(0,1),d=(6,1); transform t = rotate(-45,(3,.5)); pair e = t*a,f=t*b,g=t*c,h=t*d; pair i = intersectionpoint(a--b,e--f),j=intersectionpoint(a--b,g--h),k=intersectionpoint(c--d,e--f),l=intersectionpoint(c--d,g--h); draw(a--b^^c--d^^e--f^^g--h); filldraw(i--j--l--k--cycle,blue); label("$\alpha$",i+(-.4,.15),fontsize(8)); label("$\alpha$",i+(.4,-.15),fontsize(8)); draw(j--t*j); draw(rightanglemark(j,t*j,i), linewidth(0.5)); path lbl1 = (a+(1.5,.05))--(c+(1.5,-.05)); draw(lbl1,Arrows); label("$1$",lbl1);[/asy]](http://latex.artofproblemsolving.com/b/9/2/b92b7b2728ab336aa519c33818d4f502540198b6.png)
The rhombus has a base of length 
and height of 
. Its area is the product.
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