2001 AIME II Problems/Problem 13


In quadrilateral $ABCD$, $\angle{BAD}\cong\angle{ADC}$ and $\angle{ABD}\cong\angle{BCD}$, $AB = 8$, $BD = 10$, and $BC = 6$. The length $CD$ may be written in the form $\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Solution 1

Extend $\overline{AD}$ and $\overline{BC}$ to meet at $E$. Then, since $\angle BAD = \angle ADC$ and $\angle ABD = \angle DCE$, we know that $\triangle ABD \sim \triangle DCE$. Hence $\angle ADB = \angle DEC$, and $\triangle BDE$ is isosceles. Then $BD = BE = 10$.

[asy] /* We arbitrarily set AD = x */ real x = 60^.5, anglesize = 28;  pointpen = black; pathpen = black+linewidth(0.7); pen d = linetype("6 6")+linewidth(0.7); pair A=(0,0), D=(x,0), B=IP(CR(A,8),CR(D,10)), E=(-3x/5,0), C=IP(CR(E,16),CR(D,64/5)); D(MP("A",A)--MP("B",B,NW)--MP("C",C,NW)--MP("D",D)--cycle); D(B--D); D(A--MP("E",E)--B,d); D(anglemark(D,A,B,anglesize));D(anglemark(C,D,A,anglesize));D(anglemark(A,B,D,anglesize));D(anglemark(E,C,D,anglesize));D(anglemark(A,B,D,5/4*anglesize));D(anglemark(E,C,D,5/4*anglesize)); MP("10",(B+D)/2,SW);MP("8",(A+B)/2,W);MP("6",(B+C)/2,NW); [/asy]

Using the similarity, we have:

\[\frac{AB}{BD} = \frac 8{10} = \frac{CD}{CE} = \frac{CD}{16} \Longrightarrow CD = \frac{64}5\]

The answer is $m+n = \boxed{069}$.

Extension: To Find $AD$, use Law of Cosines on $\triangle BCD$ to get $\cos(\angle BCD)=\frac{13}{20}$ Then since $\angle BCD=\angle ABD$ use Law of Cosines on $\triangle ABD$ to find $AD=2\sqrt{15}$

Solution 2

Draw a line from $B$, parallel to $\overline{AD}$, and let it meet $\overline{CD}$ at $M$. Note that $\triangle{DAB}$ is similar to $\triangle{BMC}$ by AA similarity, since $\angle{ABD}=\angle{MCB}$ and since $BM$ is parallel to $CD$ then $\angle{BMC}=\angle{ADM}=\angle{DAB}$. Now since $ADMB$ is an isosceles trapezoid, $MD=8$. By the similarity, we have $MC=AB\cdot \frac{BC}{BD}=8\cdot \frac{6}{10}=\frac{24}{5}$, hence $CD=MC+MD=\frac{24}{5}+8=\frac{64}{5}\implies 64+5=\boxed{069}$.

Solution 3

Since $\angle{BAD}=\angle{ADM}$, if we extend AB and DC, they must meet at one point to form a isosceles triangle $\triangle{ADM}$.Now, since the problem told that $\angle{ABD}=\angle{BCD}$, we can imply that $\angle{DBM}=\angle{BCM}$ Since $\angle{M}=\angle{M}$, so $\triangle{CBM}\sim\triangle{BDM}$. Assume the length of $BM=x$;Since $\frac{BC}{MB}=\frac{DB}{MD}$ we can get $\frac{6}{x}=\frac{10}{8+x}$, we get that $x=12$.So $AM=DM=20$ similarly, we use the same pair of similar triangle we get $\frac{CM}{BM}=\frac{BM}{DM}$, we get that $CM=\frac{36}{5}$. Finally, $CD=MD-MC=\frac{64}{5}\implies 64+5=69=\boxed{069}$ ~bluesoul

Solution 3

Denote $\angle{BAD}=\angle{CDA}=x$, and $\angle{ABD}=\angle{BCD}=y$. Note that $\angle{ADB}=180^\circ-x-y$, and $\angle{DBC}=360^\circ-2x-2y$. This motivates us to draw the angle bisector of $\angle{DBC}$ because $\angle{DBC} = 2 \angle{ADB}$, so we do so and consider the intersection with $CD$ as $E$. By the angle bisector theorem, we have $\frac{CE}{DE} = \frac{BC}{BD} = \frac{3}{5}$, so we write $CE=3z$ and $DE=5z$. We also know that $\angle{EBC}=\angle{ADB}$ and $\angle{BCE}=\angle{DBA}$, so $\triangle{ADB} \sim \triangle{EBC}$. Hence, $\frac{CE}{BC}=\frac{AB}{BD}$, so we have $3z=\frac{24}{5}$. As $CD=8z$, it must be that $CD=\frac{64}{5}$, so the final answer is $\boxed{069}$.

Video Solution by OmegaLearn


~ pi_is_3.14

See also

2001 AIME II (ProblemsAnswer KeyResources)
Preceded by
Problem 12
Followed by
Problem 14
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All AIME Problems and Solutions

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