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Difference between revisions of "2009 AMC 10B Problems/Problem 20"

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== Solution 3 ==
 
== Solution 3 ==
 +
<asy>
 +
unitsize(2cm);
 +
defaultpen(linewidth(.8pt)+fontsize(8pt));
 +
dotfactor=4;
 +
 +
pair A=(0,1), B=(0,0), C=(2,0);
 +
pair D=extension(A,bisectorpoint(B,A,C),B,C);
 +
pair[] ds={A,B,C,D};
 +
 +
dot(ds);
 +
draw(A--B--C--A--D);
 +
 +
label("$1$",midpoint(A--B),W);
 +
label("$B$",B,SW);
 +
label("$D$",D,S);
 +
label("$C$",C,SE);
 +
label("$A$",A,NW);
 +
draw(rightanglemark(C,B,A,2));
 +
</asy>
  
 
== See Also ==
 
== See Also ==

Revision as of 19:05, 20 December 2020

Problem

Triangle $ABC$ has a right angle at $B$, $AB=1$, and $BC=2$. The bisector of $\angle BAC$ meets $\overline{BC}$ at $D$. What is $BD$?

[asy] unitsize(2cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4; pair A=(0,1), B=(0,0), C=(2,0); pair D=extension(A,bisectorpoint(B,A,C),B,C); pair[] ds={A,B,C,D}; dot(ds); draw(A--B--C--A--D); label("$1$",midpoint(A--B),W); label("$B$",B,SW); label("$D$",D,S); label("$C$",C,SE); label("$A$",A,NW); draw(rightanglemark(C,B,A,2)); [/asy]

$\text{(A) } \frac {\sqrt3 - 1}{2} \qquad \text{(B) } \frac {\sqrt5 - 1}{2} \qquad \text{(C) } \frac {\sqrt5 + 1}{2} \qquad \text{(D) } \frac {\sqrt6 + \sqrt2}{2} \qquad \text{(E) } 2\sqrt 3 - 1$

Solution 1

By the Pythagorean Theorem, $AC=\sqrt5$. Then, from the Angle Bisector Theorem, we have:

$\frac{BD}{1}=\frac{2-BD}{\sqrt5}\\ BD\left(1+\frac{1}{\sqrt5}\right)=\frac{2}{\sqrt5}\\ BD(\sqrt5+1)=2\\ BD=\frac{2}{\sqrt5+1}=\boxed{\frac{\sqrt5-1}{2} \implies B}.$

Solution 2

Let $\theta = \angle BAD = \angle DAC$. Notice $\tan \theta = BD$ and $\tan 2 \theta = 2$. By the double angle identity, \[2 = \frac{2 BD}{1 - BD^2} \implies BD = \boxed{\frac{\sqrt5 - 1}{2} \implies B}.\]

Solution 3

[asy] unitsize(2cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4; pair A=(0,1), B=(0,0), C=(2,0); pair D=extension(A,bisectorpoint(B,A,C),B,C); pair[] ds={A,B,C,D}; dot(ds); draw(A--B--C--A--D); label("$1$",midpoint(A--B),W); label("$B$",B,SW); label("$D$",D,S); label("$C$",C,SE); label("$A$",A,NW); draw(rightanglemark(C,B,A,2)); [/asy]

See Also

2009 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

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