Difference between revisions of "2000 AMC 10 Problems/Problem 16"
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==Problem== | ==Problem== | ||
| + | |||
| + | The diagram shows <math>28</math> lattice points, each one unit from its nearest neighbors. Segment <math>AB</math> meets segment <math>CD</math> at <math>E</math>. Find the length of segment <math>AE</math>. | ||
| + | |||
| + | <asy> | ||
| + | path seg1, seg2; | ||
| + | seg1=(6,0)--(0,3); | ||
| + | seg2=(2,0)--(4,2); | ||
| + | dot((0,0)); | ||
| + | dot((1,0)); | ||
| + | fill(circle((2,0),0.1),black); | ||
| + | dot((3,0)); | ||
| + | dot((4,0)); | ||
| + | dot((5,0)); | ||
| + | fill(circle((6,0),0.1),black); | ||
| + | dot((0,1)); | ||
| + | dot((1,1)); | ||
| + | dot((2,1)); | ||
| + | dot((3,1)); | ||
| + | dot((4,1)); | ||
| + | dot((5,1)); | ||
| + | dot((6,1)); | ||
| + | dot((0,2)); | ||
| + | dot((1,2)); | ||
| + | dot((2,2)); | ||
| + | dot((3,2)); | ||
| + | fill(circle((4,2),0.1),black); | ||
| + | dot((5,2)); | ||
| + | dot((6,2)); | ||
| + | fill(circle((0,3),0.1),black); | ||
| + | dot((1,3)); | ||
| + | dot((2,3)); | ||
| + | dot((3,3)); | ||
| + | dot((4,3)); | ||
| + | dot((5,3)); | ||
| + | dot((6,3)); | ||
| + | draw(seg1); | ||
| + | draw(seg2); | ||
| + | pair [] x=intersectionpoints(seg1,seg2); | ||
| + | fill(circle(x[0],0.1),black); | ||
| + | label("$A$",(0,3),NW); | ||
| + | label("$B$",(6,0),SE); | ||
| + | label("$C$",(4,2),NE); | ||
| + | label("$D$",(2,0),S); | ||
| + | label("$E$",x[0],N); | ||
| + | </asy> | ||
| + | |||
| + | <math>\mathrm{(A)}\ \frac{4\sqrt{5}}{3} \qquad\mathrm{(B)}\ \frac{5\sqrt{5}}{3} \qquad\mathrm{(C)}\ \frac{12\sqrt{5}}{7} \qquad\mathrm{(D)}\ 2\sqrt{5} \qquad\mathrm{(E)}\ \frac{5\sqrt{65}}{9}</math> | ||
==Solution== | ==Solution== | ||
Revision as of 00:32, 11 January 2009
Problem
The diagram shows 
lattice points, each one unit from its nearest neighbors. Segment 
meets segment 
at 
. Find the length of segment 
.
![[asy] path seg1, seg2; seg1=(6,0)--(0,3); seg2=(2,0)--(4,2); dot((0,0)); dot((1,0)); fill(circle((2,0),0.1),black); dot((3,0)); dot((4,0)); dot((5,0)); fill(circle((6,0),0.1),black); dot((0,1)); dot((1,1)); dot((2,1)); dot((3,1)); dot((4,1)); dot((5,1)); dot((6,1)); dot((0,2)); dot((1,2)); dot((2,2)); dot((3,2)); fill(circle((4,2),0.1),black); dot((5,2)); dot((6,2)); fill(circle((0,3),0.1),black); dot((1,3)); dot((2,3)); dot((3,3)); dot((4,3)); dot((5,3)); dot((6,3)); draw(seg1); draw(seg2); pair [] x=intersectionpoints(seg1,seg2); fill(circle(x[0],0.1),black); label("$A$",(0,3),NW); label("$B$",(6,0),SE); label("$C$",(4,2),NE); label("$D$",(2,0),S); label("$E$",x[0],N); [/asy]](http://latex.artofproblemsolving.com/4/1/a/41a4d56674e110de9b320f5c056a1c32c13fc24a.png)
Solution
See Also
| 2000 AMC 10 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 15 |
Followed by Problem 17 | |
| 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 | ||
