Difference between revisions of "2019 AMC 12B Problems/Problem 12"

(Problem)
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==Problem==
 
==Problem==
Right triangle <math>ACD</math> with right angle at <math>C</math> is constructed outward on the hypotenuse <math>AC</math> of isosceles right triangle <math>ABC</math> with leg length 1, as shown, so that the two triangles have equal perimeters. What is <math>\sin(2 BAD)</math>?
 
  
!! Someone with good picture-drawing skills please help !!
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Right triangle <math>ACD</math> with right angle at <math>C</math> is constructed outwards on the hypotenuse <math>\overline{AC}</math> of isosceles right triangle <math>ABC</math> with leg length <math>1</math>, as shown, so that the two triangles have equal perimeters. What is <math>\sin(2\angle BAD)</math>?
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<asy>
 +
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
 +
import graph; size(8.016233639805293cm);
 +
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 = -4.001920114613276, xmax = 4.014313525192017, ymin = -2.552570341575814, ymax = 5.6249093771911145;  /* image dimensions */
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 +
 
 +
draw((-1.6742337260757447,-1.)--(-1.6742337260757445,-0.6742337260757447)--(-2.,-0.6742337260757447)--(-2.,-1.)--cycle, linewidth(2.));
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draw((-1.7696484586262846,2.7696484586262846)--(-1.5392969172525692,3.)--(-1.7696484586262846,3.2303515413737154)--(-2.,3.)--cycle, linewidth(2.));
 +
/* draw figures */
 +
draw((-2.,3.)--(-2.,-1.), linewidth(2.));
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draw((-2.,-1.)--(2.,-1.), linewidth(2.));
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draw((2.,-1.)--(-2.,3.), linewidth(2.));
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draw((-0.6404058554606791,4.3595941445393205)--(-2.,3.), linewidth(2.));
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draw((-0.6404058554606791,4.3595941445393205)--(2.,-1.), linewidth(2.));
 +
label("$D$",(-0.9382446143428628,4.887784444795223),SE*labelscalefactor,fontsize(14));
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label("$A$",(1.9411496528285788,-1.0783204767840298),SE*labelscalefactor,fontsize(14));
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label("$B$",(-2.5046350956841272,-0.9861798602345433),SE*labelscalefactor,fontsize(14));
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label("$C$",(-2.5737405580962416,3.5747806589650395),SE*labelscalefactor,fontsize(14));
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label("$1$",(-2.665881174645728,1.2712652452278765),SE*labelscalefactor,fontsize(14));
 +
label("$1$",(-0.3393306067712029,-1.3547423264324894),SE*labelscalefactor,fontsize(14));
 +
/* dots and labels */
 +
dot((-2.,3.),linewidth(4.pt) + dotstyle);
 +
dot((-2.,-1.),linewidth(4.pt) + dotstyle);
 +
dot((2.,-1.),linewidth(4.pt) + dotstyle);
 +
dot((-0.6404058554606791,4.3595941445393205),linewidth(4.pt) + dotstyle);
 +
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
 +
/* end of picture */
 +
</asy>
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<math>\textbf{(A) } \dfrac{1}{3}  \qquad\textbf{(B) } \dfrac{\sqrt{2}}{2} \qquad\textbf{(C) } \dfrac{3}{4} \qquad\textbf{(D) } \dfrac{7}{9} \qquad\textbf{(E) }  \dfrac{\sqrt{3}}{2}</math>
  
<math>\textbf{(A) } \frac{1}{3} \qquad\textbf{(B) } \frac{\sqrt{2}}{2} \qquad\textbf{(C) } \frac{3}{4} \qquad\textbf{(D) } \frac{7}{9} \qquad\textbf{(E) } \frac{\sqrt{3}}{2}</math>
 
  
 
==Solution 1==
 
==Solution 1==

Revision as of 17:38, 14 February 2019

Problem

Right triangle $ACD$ with right angle at $C$ is constructed outwards on the hypotenuse $\overline{AC}$ of isosceles right triangle $ABC$ with leg length $1$, as shown, so that the two triangles have equal perimeters. What is $\sin(2\angle BAD)$? [asy]  /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(8.016233639805293cm);  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 = -4.001920114613276, xmax = 4.014313525192017, ymin = -2.552570341575814, ymax = 5.6249093771911145;  /* image dimensions */   draw((-1.6742337260757447,-1.)--(-1.6742337260757445,-0.6742337260757447)--(-2.,-0.6742337260757447)--(-2.,-1.)--cycle, linewidth(2.));  draw((-1.7696484586262846,2.7696484586262846)--(-1.5392969172525692,3.)--(-1.7696484586262846,3.2303515413737154)--(-2.,3.)--cycle, linewidth(2.));   /* draw figures */ draw((-2.,3.)--(-2.,-1.), linewidth(2.));  draw((-2.,-1.)--(2.,-1.), linewidth(2.));  draw((2.,-1.)--(-2.,3.), linewidth(2.));  draw((-0.6404058554606791,4.3595941445393205)--(-2.,3.), linewidth(2.));  draw((-0.6404058554606791,4.3595941445393205)--(2.,-1.), linewidth(2.));  label("$D$",(-0.9382446143428628,4.887784444795223),SE*labelscalefactor,fontsize(14));  label("$A$",(1.9411496528285788,-1.0783204767840298),SE*labelscalefactor,fontsize(14));  label("$B$",(-2.5046350956841272,-0.9861798602345433),SE*labelscalefactor,fontsize(14));  label("$C$",(-2.5737405580962416,3.5747806589650395),SE*labelscalefactor,fontsize(14));  label("$1$",(-2.665881174645728,1.2712652452278765),SE*labelscalefactor,fontsize(14));  label("$1$",(-0.3393306067712029,-1.3547423264324894),SE*labelscalefactor,fontsize(14));   /* dots and labels */ dot((-2.,3.),linewidth(4.pt) + dotstyle);  dot((-2.,-1.),linewidth(4.pt) + dotstyle);  dot((2.,-1.),linewidth(4.pt) + dotstyle);  dot((-0.6404058554606791,4.3595941445393205),linewidth(4.pt) + dotstyle);  clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);   /* end of picture */ [/asy]

$\textbf{(A) } \dfrac{1}{3}  \qquad\textbf{(B) } \dfrac{\sqrt{2}}{2} \qquad\textbf{(C) } \dfrac{3}{4} \qquad\textbf{(D) } \dfrac{7}{9} \qquad\textbf{(E) }  \dfrac{\sqrt{3}}{2}$


Solution 1

Observe that the "equal perimeter" part implies that $BC + BA = 2 = CD + DA$. A quick Pythagorean chase gives $CD = \frac{1}{2}, DA = \frac{3}{2}$. Use the sine addition formula on angles $BAC$ and $CAD$ (which requires finding their cosines as well), and this gives the sine of $BAD$. Now, use $\sin{2x} = 2\sin{x}\cos{x}$ on angle $BAD$ to get $\boxed{\textbf{(D)} = \frac{7}{9}}$.

Feel free to elaborate if necessary.

Solution 2

D 7/9 (SuperWill)

See Also

2019 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 11
Followed by
Problem 13
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All AMC 12 Problems and Solutions
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