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

(Solution 3)
(Solution 3)
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Let <math>\theta = \angle BAD = \angle DAC</math>. Notice <math>\tan \theta = BD</math> and <math>\tan 2 \theta = 2</math>. By the double angle identity, <cmath>2 = \frac{2 BD}{1 - BD^2} \implies BD = \boxed{\frac{\sqrt5 - 1}{2} \implies B}.</cmath>
 
Let <math>\theta = \angle BAD = \angle DAC</math>. Notice <math>\tan \theta = BD</math> and <math>\tan 2 \theta = 2</math>. By the double angle identity, <cmath>2 = \frac{2 BD}{1 - BD^2} \implies BD = \boxed{\frac{\sqrt5 - 1}{2} \implies B}.</cmath>
 
== 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 20:06, 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, \[2 = \frac{2 BD}{1 - BD^2} \implies BD = \boxed{\frac{\sqrt5 - 1}{2} \implies B}.\]

See Also

2009 AMC 10B (ProblemsAnswer KeyResources)
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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