Difference between revisions of "2008 AIME II Problems/Problem 5"

Problem 5

In trapezoid $ABCD$ with $\overline{BC}\parallel\overline{AD}$, let $BC = 1000$ and $AD = 2008$. Let $\angle A = 37^\circ$, $\angle D = 53^\circ$, and $M$ and $N$ be the midpoints of $\overline{BC}$ and $\overline{AD}$, respectively. Find the length $MN$.

Solution

Solution 1

Extend $\overline{AD}$ and $\overline{BC}$ to meet at a point $E$. Then $\angle AED = 180 - 53 - 37 = 90^{\circ}$.

$[asy] size(220); defaultpen(0.7+fontsize(10)); real f=100, r=1004/f; pair A=(0,0), D=(2*r, 0), N=(r,0), E=N+r*expi(74*pi/180); pair B=(126*A+125*E)/251, C=(126*D + 125*E)/251; pair[] M = intersectionpoints(N--E,B--C); draw(A--B--C--D--cycle); draw(B--E--C,dashed); draw(M[0]--N); draw(N--E,dashed); draw(rightanglemark(D,E,A,2)); picture p = new picture; draw(p,Circle(N,r),dashed+linewidth(0.5)); clip(p,A--D--D+(0,20)--A+(0,20)--cycle); add(p); label("$$A$$",A,SW); label("$$B$$",B,NW); label("$$C$$",C,NE); label("$$D$$",D,SE); label("$$E$$",E,NE); label("$$M$$",M[0],SW); label("$$N$$",N,S); label("$$1004$$",(N+D)/2,S); label("$$500$$",(M[0]+C)/2,S); [/asy]$

Since $\overline{BC} \parallel \overline{AD}$, then $BC$ and $AD$ are homothetic with respect to point $E$ by a ratio of $\frac{BC}{AD} = \frac{125}{251}$. Since the homothety carries the midpoint of $\overline{BC}$, $M$, to the midpoint of $\overline{AD}$, which is $N$, then $E,M,N$ are collinear.

As $\angle AED = 90^{\circ}$, note that the midpoint of $\overline{AD}$, $N$, is the center of the circumcircle of $\triangle AED$. We can do the same with the circumcircle about $\triangle BEC$ and $M$ (or we could apply the homothety to find $ME$ in terms of $NE$). It follows that $$NE = ND = \frac {AD}{2} = 1004, \quad ME = MC = \frac {BC}{2} = 500$$ Thus $MN = NE - ME = \boxed{504}$.

Solution 2

size(220);
defaultpen(0.7+fontsize(10));
real f=100, r=1004/f;
pair A=(0,0), D=(2*r, 0), N=(r,0), E=N+r*expi(74*pi/180);
pair B=(126*A+125*E)/251, C=(126*D + 125*E)/251;
pair[] M = intersectionpoints(N--E,B--C);
pair F = foot(B,A,D), G=foot(C,A,D), H=foot(M[0],A,D);
draw(A--B--C--D--cycle);
draw(M[0]--N); draw(B--F,dashed); draw(C--G,dashed); draw(M[0]--H,dashed);
label("$$A$$",A,SW);
label("$$B$$",B,NW);
label("$$C$$",C,NE);
label("$$D$$",D,NE);
label("$$F$$",F,S);
label("$$G$$",G,SW);
label("$$M$$",M[0],SW);
label("$$N$$",N,S);
label("$$H$$",H,S);
label("$$x$$",(N+H)/2,S);
label("$$h$$",(B+F)/2,W);
label("$$h$$",(C+G)/2,W);
label("$$1000$$",(B+C)/2,NE);
label("$$504-x$$",(G+D)/2,S);
label("$$504+x$$",(A+F)/2,S);
label("$$h$$",(M+H)/2,W);
(Error compiling LaTeX. ab3a10dd326cd803473a1100f8230fa64a9c7853.asy: 27.6: no matching function 'label(string, pair[], pair)')

Let $F,G,H$ be the feet of the perpendiculars from $B,C,M$ onto $\overline{AD}$, respectively. Let $x = NH$, so $DG = 1004 - 500 - x = 504 - x$ and $AF = 1004 - (504 - x) = 504 + x$. Also, let $h = BF = CG = HM$.

By AA~, we have that $\triangle AFB \sim \triangle CGD$, and so $$\frac{BF}{AF} = \frac {DG}{CG} \Longleftrightarrow \frac{h}{504+x} = \frac{504-x}{h} \Longrightarrow h^2 = (504 + x)(504 - x)\Longrightarrow x^2 + h^2 = 504^2.$$

By the Pythagorean Theorem on $\triangle MHN$, $$MN^{2} = x^2 + h^2 = 504^2,$$ so $MN = \boxed{504}$.