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

Latest revision as of 23:23, 20 August 2024

Problem

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 $\overline{AB}$ and $\overline{CD}$ 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]$

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 $MN = NE - ME = \boxed{504}$ .

For purposes of rigor we will show that $E,M,N$ are collinear. 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.

Solution 2

$[asy] size(220); defaultpen(0.7+fontsize(10)); real f=100, r=1004/f; pair A=(0,0), D=(2*r, 0), N=(r,0), E=N+r*expi(74*pi/180); pair B=(126*A+125*E)/251, C=(126*D + 125*E)/251; pair[] M = intersectionpoints(N--E,B--C); pair F = foot(B,A,D), G=foot(C,A,D), H=foot(M[0],A,D); draw(A--B--C--D--cycle); draw(M[0]--N); draw(B--F,dashed); draw(C--G,dashed); draw(M[0]--H,dashed); label("$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$",(N+H)/2+(0,1),S); label("$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[0]+H)/2,(1,0)); [/asy]$

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 $AF = 1004 - (500 - x) = 504 + x$ . Also, let $h = BF = CG = HM$ .

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

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

Solution 3

If you drop perpendiculars from $B$ and $C$ to $AD$ , and call the points where they meet $\overline{AD}$ , $E$ and $F$ respectively, then $FD = x$ and $EA = 1008-x$ , and so you can solve an equation in tangents. Since $\angle{A} = 37$ and $\angle{D} = 53$ , you can solve the equation [by cross-multiplication]:

$\begin{align*}\tan{37}\times (1008-x) &= \tan{53} \times x\\ \frac{(1008-x)}{x} &= \frac{\tan{53}}{\tan{37}} = \frac{\sin{53}}{\cos{53}} \times\frac{\cos{37}}{\sin{37}}\end{align*}$

However, we know that $\cos{90-x} = \sin{x}$ and $\sin{90-x} = \cos{x}$ are co-functions. Applying this,

$\begin{align*}\frac{(1008-x)}{x} &= \frac{\sin^2{53}}{\cos^2{53}} \\ x\sin^2{53} &= 1008\cos^2{53} - x\cos^2{53}\\ x(\sin^2{53} + \cos^2{53}) &= 1008\cos^2{53}\\ x = 1008\cos^2{53} &\Longrightarrow 1008-x = 1008\sin^2{53} \end{align*}$ Now, if we can find $1004 - (EA + 500)$ , and the height of the trapezoid, we can create a right triangle and use the Pythagorean Theorem to find $MN$ .

The leg of the right triangle along the horizontal is:

$1004 - 1008\sin^2{53} - 500 = 504 - 1008\sin^2{53}.$

Now to find the other leg of the right triangle (also the height of the trapezoid), we can simplify the following expression:

$\begin{align*}\tan{37} \times 1008 \sin^2{53} = \tan{37} \times 1008 \cos^2{37} = 1008\cos{37}\sin{37} = 504\sin74\end{align*}$

Now we used Pythagorean Theorem and get that $MN$ is equal to:

$\begin{align*}&\sqrt{(1008\sin^2{53} + 500 -1004)^2 + (504\sin{74})^2} = 504\sqrt{1-2\sin^2{53} + \sin^2{74}} \end{align*}$

However, $1-2\sin^2{53} = \cos^2{106}$ and $\sin^2{74} = \sin^2{106}$ so now we end up with:

$504\sqrt{\cos^2{106} + \sin^2{106}} =\fbox{504}.$

Solution 4

Plot the trapezoid such that $B=\left(1000\cos 37^\circ, 0\right)$ , $C=\left(0, 1000\sin 37^\circ\right)$ , $A=\left(2008\cos 37^\circ, 0\right)$ , and $D=\left(0, 2008\sin 37^\circ\right)$ .

The midpoints of the requested sides are $\left(500\cos 37^\circ, 500\sin 37^\circ\right)$ and $\left(1004\cos 37^\circ, 1004\sin 37^\circ\right)$ .

To find the distance from $M$ to $N$ , we simply apply the distance formula and the Pythagorean identity $\sin^2 x + \cos^2 x = 1$ to get $MN=\boxed{504}$ .

Solution 5

Similar to solution 1; Notice that it forms a right triangle. Remembering that the median to the hypotenuse is simply half the length of the hypotenuse, we quickly see that the length is $\frac{2008}{2}-\frac{1000}{2}=504$ .

Solution 6

Obviously, these angles are random--the only special thing about them is that they add up to 90. So we might as well let the given angles equal 45 and 45, and now the answer is trivially $\boxed{504}$ . (The trapezoid is isosceles, and you see two 45-45-90 triangles;from there you can get the answer.)

Solution 7

Let the height be h. Note that if $\overline{NH} = x$ then if we draw perpendiculars like in solution 1, $\overline{FN} = 500 - x, \overline{AF} = 504 + x, \overline{HG} = 500, \overline{GD} = 504 - x.$ Note that we wish to find $\overline{MN} = \sqrt{x^2 + h^2}.$ Let's find $\tan(53)$ in two ways. Finding $\tan(53)$ from $\triangle BAF$ yields $\tan(53) = \frac{504+x}{h}.$ Finding it from $\triangle CDG$ yields $\frac{h}{504-x}.$ Setting these equal yields $\frac{504+x}{h}=\frac{h}{504-x} \rightarrow h^2 = 504^2-x^2 \rightarrow \sqrt{x^2+h^2} = \sqrt{504^2} = \boxed{504}$

Solution 8

Rotate trapezoid $MNCD$ 180 degrees around point $N$ so that $AN$ coincides with $ND$ . Let the image of trapezoid $MNCD$ be $ANM'C'$ . Since angles are preserved during rotations, $\angle BAC' = 37^{\circ} + 53 ^{\circ} =90 ^{\circ}$ . Since $BM=CM=C'M'$ and $BM || C'M'$ , $BMM'C'$ is a parallelogram. Thus, $MM'=BC'$ . Let the point where $BC'$ intersects $AD$ be $E$ . Since $BMNE$ is a parallelogram, $AE=AN-BM=1004-500-504$ . Since $BE=EC$ and $\angle BAC= 90^{\circ}$ , $AE$ is a median to the hypotenuse of $BAC'$ . Therefore, $BC'=2 AE= 1008$ , and $BE=MN=\boxed{504}.$

@@ Line 1: / Line 1: @@
-== Problem 5 ==
+__TOC__
+== Problem==
 In [[trapezoid]] <math>ABCD</math> with <math>\overline{BC}\parallel\overline{AD}</math>, let <math>BC = 1000</math> and <math>AD = 2008</math>. Let <math>\angle A = 37^\circ</math>, <math>\angle D = 53^\circ</math>, and <math>M</math> and <math>N</math> be the [[midpoint]]s of <math>\overline{BC}</math> and <math>\overline{AD}</math>, respectively. Find the length <math>MN</math>.
-__TOC__
 == Solution ==
 === Solution 1 ===
-Extend <math>\overline{AD}</math> and <math>\overline{BC}</math> to meet at a point <math>E</math>. Then <math>\angle AED = 180 - 53 - 37 = 90^{\circ}</math>.
+Extend <math>\overline{AB}</math> and <math>\overline{CD}</math> to meet at a point <math>E</math>. Then <math>\angle AED = 180 - 53 - 37 = 90^{\circ}</math>.
 <center><asy>
 size(220);
@@ Line 31: / Line 32: @@
 label("\(500\)",(M[0]+C)/2,S);
 </asy></center>
-Since <math>\overline{BC} \parallel \overline{AD}</math>, then <math>BC</math> and <math>AD</math> are [[homothecy|homothetic]] with respect to point <math>E</math> by a ratio of <math>\frac{BC}{AD} = \frac{125}{251}</math>. Since the homothety carries the midpoint of <math>\overline{BC}</math>, <math>M</math>, to the midpoint of <math>\overline{AD}</math>, which is <math>N</math>, then <math>E,M,N</math> are [[collinear]].
+As <math>\angle AED = 90^{\circ}</math>, note that the midpoint of <math>\overline{AD}</math>, <math>N</math>, is the center of the [[circumcircle]] of <math>\triangle AED</math>. We can do the same with the circumcircle about <math>\triangle BEC</math> and <math>M</math> (or we could apply the homothety to find <math>ME</math> in terms of <math>NE</math>). It follows that
+<cmath>NE = ND = \frac {AD}{2} = 1004, \quad ME = MC = \frac {BC}{2} = 500.</cmath>
-As <math>\angle AED = 90^{\circ}</math>, note that the midpoint of <math>\overline{AD}</math>, <math>N</math>, is the center of the [[circumcircle]] of <math>\triangle AED</math>. We can do the same with the circumcircle about <math>\triangle BEC</math> and <math>M</math> (or we could apply the homothety to find <math>ME</math> in terms of <math>NE</math>). It follows that
-<cmath>
-NE = ND = \frac {AD}{2} = 1004, \quad ME = MC = \frac {BC}{2} = 500
-</cmath>
 Thus <math>MN = NE - ME = \boxed{504}</math>.
+For purposes of rigor we will show that <math>E,M,N</math> are collinear. Since <math>\overline{BC} \parallel \overline{AD}</math>, then <math>BC</math> and <math>AD</math> are [[homothecy|homothetic]] with respect to point <math>E</math> by a ratio of <math>\frac{BC}{AD} = \frac{125}{251}</math>. Since the homothety carries the midpoint of <math>\overline{BC}</math>, <math>M</math>, to the midpoint of <math>\overline{AD}</math>, which is <math>N</math>, then <math>E,M,N</math> are [[collinear]].
 === Solution 2 ===
@@ Line 65: / Line 67: @@
 label("\(504-x\)",(G+D)/2,S);
 label("\(504+x\)",(A+F)/2,S);
-label("\(h\)",(M+H)/2,W);
+label("\(h\)",(M[0]+H)/2,(1,0));
 </asy></center>
-Let <math>F,G,H</math> be the feet of the [[perpendicular]]s from <math>B,C,M</math> onto <math>\overline{AD}</math>, respectively. Let <math>x = NH</math>, so <math>DG = 1004 - 500 - x = 504 - x</math> and <math>AF = 1004 - (504 - x) = 504 + x</math>. Also, let <math>h = BF = CG = HM</math>.
+Let <math>F,G,H</math> be the feet of the [[perpendicular]]s from <math>B,C,M</math> onto <math>\overline{AD}</math>, respectively. Let <math>x = NH</math>, so <math>DG = 1004 - 500 - x = 504 - x</math> and <math>AF = 1004 - (500 - x) = 504 + x</math>. Also, let <math>h = BF = CG = HM</math>.
 By AA~, we have that <math>\triangle AFB \sim \triangle CGD</math>, and so
@@ Line 76: / Line 78: @@
 === Solution 3 ===
-If you drop perpendiculars from B and C to AD, and call the points if you drop perpendiculars from B and C to AD and call the points where they meet AD E and F respectively and call FD = x and EA = <math>1008-x</math> , then you can solve an equation in tangents. Since <math>\angle{A} = 37</math> and <math>\angle{D} = 53</math>, you can solve the equation:
+If you drop perpendiculars from <math>B</math> and <math>C</math> to <math>AD</math>, and call the points where they meet <math>\overline{AD}</math>, <math>E</math> and <math>F</math> respectively, then <math>FD = x</math> and <math>EA = 1008-x</math> , and so you can solve an equation in tangents. Since <math>\angle{A} = 37</math> and <math>\angle{D} = 53</math>, you can solve the equation [by cross-multiplication]:
-<math>\tan{37}\times (1008-x) = \tan{53} \times x</math>.
+<cmath>\begin{align*}\tan{37}\times (1008-x) &= \tan{53} \times x\\
+\frac{(1008-x)}{x} &= \frac{\tan{53}}{\tan{37}} = \frac{\sin{53}}{\cos{53}} \times\frac{\cos{37}}{\sin{37}}\end{align*}</cmath>
-Now if you cross multiply, you get the equation:
+However, we know that <math>\cos{90-x} = \sin{x}</math> and <math>\sin{90-x} = \cos{x}</math> are co-functions. Applying this,
-<math>\frac{(1008-x)}{x} = \frac{\tan{53}}{\tan{37}} = \frac{\sin{53}}{\cos{53}} \times\frac{\sin{37}}{\cos{37}}</math>
+<cmath>\begin{align*}\frac{(1008-x)}{x} &= \frac{\sin^2{53}}{\cos^2{53}} \\
+x\sin^2{53} &= 1008\cos^2{53} - x\cos^2{53}\\
+x(\sin^2{53}  + \cos^2{53}) &= 1008\cos^2{53}\\
+x = 1008\cos^2{53} &\Longrightarrow 1008-x = 1008\sin^2{53}
+\end{align*}</cmath>
+Now, if we can find <math>1004 - (EA + 500)</math>, and the height of the trapezoid, we can create a right triangle and use the [[Pythagorean Theorem]] to find <math>MN</math>.
-However, we know that <math>\cos{90-x} = sin{x}</math> and <math>\sin{90-x} = cos{x}</math>. So if we apply that, we end up with the equation:
+The leg of the right triangle along the horizontal is:
-<math>\frac{(1008-x)}{x} = \frac{\sin^2{53}}{\cos^2{53}}</math>.
+<cmath>1004 - 1008\sin^2{53} - 500 = 504 - 1008\sin^2{53}.</cmath>
-so if we cross multiply again, we get:
+Now to find the other leg of the right triangle (also the height of the trapezoid), we can simplify the following expression:
-<math>x\sin^2{53} = 1008\cos^2{53} - x\cos^2{53}</math>
+<cmath>\begin{align*}\tan{37} \times 1008 \sin^2{53}
-<math>x(\sin^2{53}  + \cos^2{53}) = 1008\cos^2{53}</math>
+= \tan{37} \times 1008 \cos^2{37}
-<math>x = 1008\cos^2{53}</math>,
+= 1008\cos{37}\sin{37}
-<math>1008-x = 1008\sin^2{53}</math>.
+= 504\sin74\end{align*}</cmath>
-Now, if we can find <math>1004 - (EA + 500)</math>, and the height of the trapezoid, we can create a right triangle and  use the Pythagorean Theorem to find <math>MN</math>.
+Now we used Pythagorean Theorem and get that <math>MN</math> is equal to:
-The leg of the right triangle along the horizontal is:
+<cmath>
+\begin{align*}&\sqrt{(1008\sin^2{53} + 500 -1004)^2 + (504\sin{74})^2} = 504\sqrt{1-2\sin^2{53} + \sin^2{74}}
+\end{align*}
+</cmath>
-<math>1004 - 1008\sin^2{53} - 500 = 504 - 1008\sin^2{53}</math>.
+However, <math>1-2\sin^2{53} = \cos^2{106}</math> and <math>\sin^2{74} = \sin^2{106}</math> so now we end up with:
-Now to find the other leg of the right triangle (also the height of the trapezoid), we can simplify the following expression:
+<cmath>504\sqrt{\cos^2{106} + \sin^2{106}} =\fbox{504}.</cmath>
-<math>\tan{37} \times 1008 \sin^2{53}</math>
+=== Solution 4 ===
-=<math>\tan{37} \times 1008 \cos^2{37}</math>
-=<math>1008\cos{37}\sin{37}</math>
-=<math>504\sin74</math>
-Now we used Pythagorean Theorem and get that MN is equal to:
+Plot the trapezoid such that <math>B=\left(1000\cos 37^\circ, 0\right)</math>, <math>C=\left(0, 1000\sin 37^\circ\right)</math>, <math>A=\left(2008\cos 37^\circ, 0\right)</math>, and <math>D=\left(0, 2008\sin 37^\circ\right)</math>.
-<math>\sqrt{(1008\sin^2{53} + 500 -1004)^2 + (504\sin{74})^2}</math>
+The midpoints of the requested sides are <math>\left(500\cos 37^\circ, 500\sin 37^\circ\right)</math> and <math>\left(1004\cos 37^\circ, 1004\sin 37^\circ\right)</math>.
-=<math>504\sqrt{1-2\sin^2{53} + \sin^2{74}}</math>
+To find the distance from <math>M</math> to <math>N</math>, we simply apply the distance formula and the Pythagorean identity <math>\sin^2 x + \cos^2 x = 1</math> to get <math>MN=\boxed{504}</math>.
-However,
-<math>1-2\sin^2{53}</math>= <math>\cos^2{106}</math>
-and <math>\sin^2{74} = \sin^2{106}</math>
+=== Solution 5 ===
-so now we end up with:
+Similar to solution 1; Notice that it forms a right triangle. Remembering that the median to the hypotenuse is simply half the length of the hypotenuse, we quickly see that the length is <math>\frac{2008}{2}-\frac{1000}{2}=504</math>.
-<math>504\sqrt{\cos^2{106} + \sin^2{106}}</math>
+=== Solution 6 ===
-=<math>504\sqrt{1}</math>
+Obviously, these angles are random--the only special thing about them is that they add up to 90. So we might as well let the given angles equal 45 and 45, and now the answer is trivially <math>\boxed{504}</math>. (The trapezoid is isosceles, and you see two 45-45-90 triangles;from there you can get the answer.)
-=<math>\fbox{504}</math>
+== Solution 7 ==
+Let the height be h. Note that if <math>\overline{NH} = x</math> then if we draw perpendiculars like in solution 1, <math>\overline{FN} = 500 - x, \overline{AF} = 504 + x, \overline{HG} = 500, \overline{GD} = 504 - x.</math> Note that we wish to find <math>\overline{MN} = \sqrt{x^2 + h^2}.</math> Let's find <math>\tan(53)</math> in two ways. Finding <math>\tan(53)</math> from <math>\triangle BAF</math> yields <math>\tan(53) = \frac{504+x}{h}.</math> Finding it from <math>\triangle CDG</math> yields <math>\frac{h}{504-x}.</math> Setting these equal yields
+<cmath>\frac{504+x}{h}=\frac{h}{504-x} \rightarrow h^2 = 504^2-x^2 \rightarrow \sqrt{x^2+h^2} = \sqrt{504^2} = \boxed{504}</cmath>
+== Solution 8 ==
+Rotate trapezoid <math>MNCD</math> 180 degrees around point <math>N</math> so that <math>AN</math> coincides with <math>ND</math>. Let the image of trapezoid <math>MNCD</math> be <math>ANM'C'</math>. Since angles are preserved during rotations, <math>\angle BAC' = 37^{\circ} + 53 ^{\circ} =90 ^{\circ}</math>. Since <math>BM=CM=C'M'</math> and <math>BM || C'M'</math>, <math>BMM'C'</math> is a parallelogram. Thus, <math>MM'=BC'</math>.
+Let the point where <math>BC'</math> intersects <math>AD</math> be <math>E</math>. Since <math>BMNE</math> is a parallelogram, <math>AE=AN-BM=1004-500-504</math>. Since <math>BE=EC</math> and <math>\angle BAC= 90^{\circ}</math>, <math>AE</math> is a median to the hypotenuse of <math>BAC'</math>. Therefore, <math>BC'=2 AE= 1008</math>, and <math>BE=MN=\boxed{504}.</math>
 == See also ==
 {{AIME box|year=2008|n=II|num-b=4|num-a=6}}
@@ Line 130: / Line 141: @@
 [[Category:Intermediate Geometry Problems]]
+{{MAA Notice}}

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