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Difference between revisions of "2004 AMC 10B Problems/Problem 24"
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Set <math>\segment BD</math>'s length as <math>x</math>. <math>CD</math>'s length must also be <math>x</math> since <math> \angle BAD </math> and <math> \angle DAC </math> intercept arcs of equal length. Using [[Ptolemy's Theorem]], <math>7x+8x=9(AD)</math>. The ratio is <math>\boxed{\frac{5}{3}}\implies(B)</math>
 
Set <math>\segment BD</math>'s length as <math>x</math>. <math>CD</math>'s length must also be <math>x</math> since <math> \angle BAD </math> and <math> \angle DAC </math> intercept arcs of equal length. Using [[Ptolemy's Theorem]], <math>7x+8x=9(AD)</math>. The ratio is <math>\boxed{\frac{5}{3}}\implies(B)</math>
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==Solution 2==
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Let <math>P = \overline{BC}\cap \overline{AD}</math>.  Observe that <math>\angle ABC = \angle ADC</math> because they subtend the same arc.  Furthermore, <math>\angle BAP = \angle PAC</math>, so <math>\triangle ABP</math> is similar to <math>\triangle ADC</math> by AAA similarity.  Then <math>\dfrac{AD}{AB} = \dfrac{CD}{BP}</math>.  By angle bisector theorem, <math>\dfrac{7}{BP} = \dfrac{8}{CP}</math> so <math>\dfrac{7}{BP} = \dfrac{8}{9-BP}</math> which gives <math>BP = \dfrac{21}{5}</math>.  Plugging this into the similarity proportion gives:  <math>\dfrac{AD}{7} = \dfrac{CD}{\dfrac{21}{5}} \implies \dfrac{AD}{CD} = \boxed{\dfrac{5}{3}} = \textbf{B}</math>.
  
 
== See Also ==
 
== See Also ==

Revision as of 16:45, 27 December 2013

In triangle $ABC$ we have $AB=7$, $AC=8$, $BC=9$. Point $D$ is on the circumscribed circle of the triangle so that $AD$ bisects angle $BAC$. What is the value of $AD/CD$?

$\text{(A) } \dfrac{9}{8} \quad \text{(B) } \dfrac{5}{3} \quad \text{(C) } 2 \quad \text{(D) } \dfrac{17}{7} \quad \text{(E) } \dfrac{5}{2}$


Solution

Set $\segment BD$ (Error compiling LaTeX. Unknown error_msg)'s length as $x$. $CD$'s length must also be $x$ since $\angle BAD$ and $\angle DAC$ intercept arcs of equal length. Using Ptolemy's Theorem, $7x+8x=9(AD)$. The ratio is $\boxed{\frac{5}{3}}\implies(B)$

Solution 2

Let $P = \overline{BC}\cap \overline{AD}$. Observe that $\angle ABC = \angle ADC$ because they subtend the same arc. Furthermore, $\angle BAP = \angle PAC$, so $\triangle ABP$ is similar to $\triangle ADC$ by AAA similarity. Then $\dfrac{AD}{AB} = \dfrac{CD}{BP}$. By angle bisector theorem, $\dfrac{7}{BP} = \dfrac{8}{CP}$ so $\dfrac{7}{BP} = \dfrac{8}{9-BP}$ which gives $BP = \dfrac{21}{5}$. Plugging this into the similarity proportion gives: $\dfrac{AD}{7} = \dfrac{CD}{\dfrac{21}{5}} \implies \dfrac{AD}{CD} = \boxed{\dfrac{5}{3}} = \textbf{B}$.

See Also

2004 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 23 		Followed by
Problem 25
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All AMC 10 Problems and Solutions

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