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Difference between revisions of "2004 AMC 10B Problems/Problem 20"

(New page: == Problem == In <math>\triangle ABC</math> points <math>D</math> and <math>E</math> lie on <math>BC</math> and <math>AC</math>, respectively. If <math>AD</math> and <math>BE</math> inter...)
 
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</asy>
 
</asy>
  
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== Solution ==
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== Solution (Mass points) ==
 +
 
 +
The presence of only ratios in the problem essentially cries out for mass points.
 +
 
 +
As per the problem, we assign a mass of <math>1</math> to point <math>A</math>, and a mass of <math>3</math> to <math>D</math>. Then, to balance <math>A</math> and <math>D</math> on <math>T</math>, <math>T</math> has a mass of <math>4</math>.
 +
 
 +
Now, were we to assign a mass of <math>1</math> to <math>B</math> and a mass of <math>4</math> to <math>E</math>, we'd have <math>5T</math>. Scaling this down by <math>4/5</math> (to get <math>4T</math>, which puts <math>B</math> and <math>E</math> in terms of the masses of <math>A</math> and <math>D</math>), we assign a mass of <math>\frac{4}{5}</math> to <math>B</math> and a mass of <math>\frac{16}{5}</math> to <math>E</math>.
 +
 
 +
Now, to balance <math>A</math> and <math>C</math> on <math>E</math>, we must give <math>C</math> a mass of <math>\frac{16}{5}-1=\frac{11}{5}</math>.
 +
 
 +
Finally, the ratio of <math>CD</math> to <math>BD</math> is given by the ratio of the mass of <math>B</math> to the mass of <math>C</math>, which is <math>\frac{4}{5}\cdot\frac{5}{11}=\boxed{\textbf{(D)}\ \frac{4}{11}}</math>.
 +
 
 +
== Solution (Coordinates) ==
  
 
Affine transformations preserve ratios of distances, and for any pair of triangles there is an affine transformation that maps the first one onto the second one. This is why the answer is the same for any <math>\triangle ABC</math>, and we just need to compute it for any single triangle.
 
Affine transformations preserve ratios of distances, and for any pair of triangles there is an affine transformation that maps the first one onto the second one. This is why the answer is the same for any <math>\triangle ABC</math>, and we just need to compute it for any single triangle.

Revision as of 18:19, 30 March 2009

Problem

In $\triangle ABC$ points $D$ and $E$ lie on $BC$ and $AC$, respectively. If $AD$ and $BE$ intersect at $T$ so that $AT/DT=3$ and $BT/ET=4$, what is $CD/BD$?


$\mathrm{(A) \ } \frac{1}{8} \qquad \mathrm{(B) \ } \frac{2}{9} \qquad \mathrm{(C) \ } \frac{3}{10} \qquad \mathrm{(D) \ } \frac{4}{11} \qquad \mathrm{(E) \ } \frac{5}{12}$


[asy] unitsize(1cm); defaultpen(0.8); pair A=(0,0), B=5*dir(60), C=5*(1,0), D=B + (11/15)*(C-B), E = A + (11/16)*(C-A); draw(A--B--C--cycle); draw(A--D); draw(B--E); pair T=intersectionpoint(A--D,B--E); label("$A$",A,SW); label("$B$",B,N); label("$C$",C,SE); label("$D$",D,NE); label("$E$",E,S); label("$T$",T,2*WNW); [/asy]

Solution (Mass points)

The presence of only ratios in the problem essentially cries out for mass points.

As per the problem, we assign a mass of $1$ to point $A$, and a mass of $3$ to $D$. Then, to balance $A$ and $D$ on $T$, $T$ has a mass of $4$.

Now, were we to assign a mass of $1$ to $B$ and a mass of $4$ to $E$, we'd have $5T$. Scaling this down by $4/5$ (to get $4T$, which puts $B$ and $E$ in terms of the masses of $A$ and $D$), we assign a mass of $\frac{4}{5}$ to $B$ and a mass of $\frac{16}{5}$ to $E$.

Now, to balance $A$ and $C$ on $E$, we must give $C$ a mass of $\frac{16}{5}-1=\frac{11}{5}$.

Finally, the ratio of $CD$ to $BD$ is given by the ratio of the mass of $B$ to the mass of $C$, which is $\frac{4}{5}\cdot\frac{5}{11}=\boxed{\textbf{(D)}\ \frac{4}{11}}$.

Solution (Coordinates)

Affine transformations preserve ratios of distances, and for any pair of triangles there is an affine transformation that maps the first one onto the second one. This is why the answer is the same for any $\triangle ABC$, and we just need to compute it for any single triangle.

We can choose the points $A=(-3,0)$, $B=(0,4)$, and $D=(1,0)$. This way we will have $T=(0,0)$, and $E=(0,-1)$. The situation is shown in the picture below:

[asy] unitsize(1cm); defaultpen(0.8); pair A=(-3,0), B=(0,4), C=(15/11,-16/11), D=(1,0), E=(0,-1); draw(A--B--C--cycle); draw(A--D); draw(B--E); pair T=intersectionpoint(A--D,B--E); label("$A$",A,SW); label("$B$",B,N); label("$C$",C,SE); label("$D$",D,NE); label("$E$",E,S); label("$T$",T,NW); label("$3$",A--T,N); label("$4$",B--T,W); label("$1$",D--T,N); label("$1$",E--T,W); [/asy]

The point $C$ is the intersection of the lines $BD$ and $AE$. The points on the first line have the form $(t,4-4t)$, the points on the second line have the form $(t,-1-t/3)$. Solving for $t$ we get $t=15/11$, hence $C=(15/11,-16/11)$.

The ratio $CD/BD$ can now be computed simply by observing the $x$ coordinates of $B$, $C$, and $D$:

\[\frac{CD}{BD} = \frac{15/11 - 1}{1 - 0} = \boxed{\frac 4{11}}\]

See also

2004 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 19 		Followed by
Problem 21
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions
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