Difference between revisions of "2014 AMC 12A Problems/Problem 20"
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==Solution 2== | ==Solution 2== | ||
+ | <asy> | ||
+ | import graph; size(20.95cm); | ||
+ | real labelscalefactor = 0.5; /* changes label-to-point distance */ | ||
+ | pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ | ||
+ | pen dotstyle = black; /* point style */ | ||
+ | real xmin = -1.52, xmax = 19.43, ymin = -2.35, ymax = 10.68; /* image dimensions */ | ||
+ | |||
+ | |||
+ | draw(arc((8.03,9.81),0.61,-124.95,-84.95)--(8.03,9.81)--cycle); | ||
+ | draw(arc((8.03,9.81),0.61,-164.95,-124.95)--(8.03,9.81)--cycle); | ||
+ | draw(arc((8.03,9.81),0.61,-84.95,-44.95)--(8.03,9.81)--cycle); | ||
+ | /* draw figures */ | ||
+ | draw((8.03,9.81)--(2.3,1.61)); | ||
+ | draw((8.03,9.81)--(8.56,3.83)); | ||
+ | draw((2.3,1.61)--(8.56,3.83)); | ||
+ | draw((8.03,9.81)--(2.24,8.25)); | ||
+ | draw((8.03,9.81)--(15.11,2.74)); | ||
+ | draw((2.24,8.25)--(15.11,2.74)); | ||
+ | /* dots and labels */ | ||
+ | |||
+ | label("$A$", (7.9,10.03), NE * labelscalefactor); | ||
+ | |||
+ | label("$B$", (1.91,1.75), NE * labelscalefactor); | ||
+ | label("$40^\circ$", (7.58,8.84), NE * labelscalefactor); | ||
+ | |||
+ | label("$C$", (8.82,3.42), NE * labelscalefactor); | ||
+ | |||
+ | label("$B'$", (15.47,2.6), NE * labelscalefactor); | ||
+ | |||
+ | label("$C'$", (1.85,8.42), NE * labelscalefactor); | ||
+ | label("$6$", (5.05,9.35), NE * labelscalefactor); | ||
+ | label("$10$", (11.45,6.7), NE * labelscalefactor); | ||
+ | |||
+ | label("$D$", (6.21,6.68), NE * labelscalefactor); | ||
+ | |||
+ | label("$E$", (7.86,6.11), NE * labelscalefactor); | ||
+ | label("$40^\circ$", (6.99,9.23), NE * labelscalefactor); | ||
+ | label("$40^\circ$", (8.23,8.86), NE * labelscalefactor); | ||
+ | clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); | ||
+ | </asy> | ||
+ | |||
+ | (Diagram by dasobson) | ||
Reflect <math>C</math> across <math>AB</math> to <math>C'</math>. Similarly, reflect <math>B</math> across <math>AC</math> to <math>B'</math>. Clearly, <math>BE = B'E</math> and <math>CD = C'D</math>. Thus, the sum <math>BE + DE + CD = B'E + DE + C'E</math>. This value is maximized when <math>B'</math>, <math>C'</math>, <math>D</math> and <math>E</math> are collinear. To finish, we use the law of cosines on the triangle <math>AB'C'</math>: <math>B'C' = \sqrt{6^2 + 10^2 - 2(6)(10)\cos 120} = 14</math> | Reflect <math>C</math> across <math>AB</math> to <math>C'</math>. Similarly, reflect <math>B</math> across <math>AC</math> to <math>B'</math>. Clearly, <math>BE = B'E</math> and <math>CD = C'D</math>. Thus, the sum <math>BE + DE + CD = B'E + DE + C'E</math>. This value is maximized when <math>B'</math>, <math>C'</math>, <math>D</math> and <math>E</math> are collinear. To finish, we use the law of cosines on the triangle <math>AB'C'</math>: <math>B'C' = \sqrt{6^2 + 10^2 - 2(6)(10)\cos 120} = 14</math> | ||
Revision as of 01:07, 27 February 2018
Contents
Problem
In , , , and . Points and lie on and respectively. What is the minimum possible value of ?
Solution 1
Let be the reflection of across , and let be the reflection of across . Then it is well-known that the quantity is minimized when it is equal to . (Proving this is a simple application of the triangle inequality; for an example of a simpler case, see Heron's Shortest Path Problem.) As lies on both and , we have . Furthermore, by the nature of the reflection, so . Therefore by the Law of Cosines
Solution 2
(Diagram by dasobson) Reflect across to . Similarly, reflect across to . Clearly, and . Thus, the sum . This value is maximized when , , and are collinear. To finish, we use the law of cosines on the triangle :
See Also
2014 AMC 12A (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 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.