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Difference between revisions of "2014 AMC 12A Problems/Problem 20"

(→‎Solution)
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==Solution==
 
==Solution==
 
Let <math>C_1</math> be the reflection of <math>C</math> across <math>\overline{AB}</math>, and let <math>C_2</math> be the reflection of <math>C_1</math> across <math>\overline{AC}</math>.  Then it is well-known that the quantity <math>BE+DE+CD</math> is minimized when it is equal to <math>C_2B</math>.  (Proving this is a simple application of the triangle inequality; for an example of a simpler case, see Heron's Shortest Path Problem.)  As <math>A</math> lies on both <math>AB</math> and <math>AC</math>, we have <math>C_2A=C_1A=CA=6</math>.  Furthermore, <math>\angle CAC_1=2\angle CAB=80^\circ</math> by the nature of the reflection, so <math>\angle C_2AB=\angle C_2AC+\angle CAB=80^\circ+40^\circ=120^\circ</math>.  Therefore by the Law of Cosines <cmath>BC_2^2=6^2+10^2-2\cdot 6\cdot 10\cos 120^\circ=196\implies BC_2=\boxed{14\textbf{ (D)}}.</cmath>
 
Let <math>C_1</math> be the reflection of <math>C</math> across <math>\overline{AB}</math>, and let <math>C_2</math> be the reflection of <math>C_1</math> across <math>\overline{AC}</math>.  Then it is well-known that the quantity <math>BE+DE+CD</math> is minimized when it is equal to <math>C_2B</math>.  (Proving this is a simple application of the triangle inequality; for an example of a simpler case, see Heron's Shortest Path Problem.)  As <math>A</math> lies on both <math>AB</math> and <math>AC</math>, we have <math>C_2A=C_1A=CA=6</math>.  Furthermore, <math>\angle CAC_1=2\angle CAB=80^\circ</math> by the nature of the reflection, so <math>\angle C_2AB=\angle C_2AC+\angle CAB=80^\circ+40^\circ=120^\circ</math>.  Therefore by the Law of Cosines <cmath>BC_2^2=6^2+10^2-2\cdot 6\cdot 10\cos 120^\circ=196\implies BC_2=\boxed{14\textbf{ (D)}}.</cmath>
 +
 +
==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>
  
 
==See Also==
 
==See Also==
 
{{AMC12 box|year=2014|ab=A|num-b=19|num-a=21}}
 
{{AMC12 box|year=2014|ab=A|num-b=19|num-a=21}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 13:15, 1 January 2018

Problem

In $\triangle BAC$, $\angle BAC=40^\circ$, $AB=10$, and $AC=6$. Points $D$ and $E$ lie on $\overline{AB}$ and $\overline{AC}$ respectively. What is the minimum possible value of $BE+DE+CD$?

$\textbf{(A) }6\sqrt 3+3\qquad \textbf{(B) }\dfrac{27}2\qquad \textbf{(C) }8\sqrt 3\qquad \textbf{(D) }14\qquad \textbf{(E) }3\sqrt 3+9\qquad$

Solution

Let $C_1$ be the reflection of $C$ across $\overline{AB}$, and let $C_2$ be the reflection of $C_1$ across $\overline{AC}$. Then it is well-known that the quantity $BE+DE+CD$ is minimized when it is equal to $C_2B$. (Proving this is a simple application of the triangle inequality; for an example of a simpler case, see Heron's Shortest Path Problem.) As $A$ lies on both $AB$ and $AC$, we have $C_2A=C_1A=CA=6$. Furthermore, $\angle CAC_1=2\angle CAB=80^\circ$ by the nature of the reflection, so $\angle C_2AB=\angle C_2AC+\angle CAB=80^\circ+40^\circ=120^\circ$. Therefore by the Law of Cosines \[BC_2^2=6^2+10^2-2\cdot 6\cdot 10\cos 120^\circ=196\implies BC_2=\boxed{14\textbf{ (D)}}.\]

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 $C$ across $AB$ to $C'$. Similarly, reflect $B$ across $AC$ to $B'$. Clearly, $BE = B'E$ and $CD = C'D$. Thus, the sum $BE + DE + CD = B'E + DE + C'E$. This value is maximized when $B'$, $C'$, $D$ and $E$ are collinear. To finish, we use the law of cosines on the triangle $AB'C'$: $B'C' = \sqrt{6^2 + 10^2 - 2(6)(10)\cos 120} = 14$

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

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