2004 AMC 10B Problems/Problem 20
Problem
In 
points 
and 
lie on 
and 
, respectively. If 
and 
intersect at 
so that 
and 
, what is 
?
![[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]](http://latex.artofproblemsolving.com/0/5/b/05b2f751804dc3a7866b1e5c5e9c3f214a502ad1.png)
Solution
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 , and we just need to compute it for any single triangle.
We can choose the points 
, 
, and 
. This way we will have 
, and 
. 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]](http://latex.artofproblemsolving.com/a/3/3/a33ad88d1e8b0bec0d13245fe244c3096f00077e.png)
The point 
is the intersection of the lines 
and 
. The points on the first line have the form 
, the points on the second line have the form 
. Solving for 
we get 
, hence 
.
The ratio can now be computed simply by observing the 
coordinates of 
, , and :
![\[\frac{CD}{BD} = \frac{15/11 - 1}{1 - 0} = \boxed{\frac 4{11}}\]](http://latex.artofproblemsolving.com/1/9/5/1957a477c93b2f66ad7b9d706b732a4940a6173f.png)
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 | ||
