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Difference between revisions of "2021 Fall AMC 12B Problems/Problem 24"

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==Problem==
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Triangle <math>ABC</math> has side lengths <math>AB = 11, BC=24</math>, and <math>CA = 20</math>. The bisector of <math>\angle{BAC}</math> intersects <math>\overline{BC}</math> in point <math>D</math>, and intersects the circumcircle of <math>\triangle{ABC}</math> in point <math>E \ne A</math>. The circumcircle of <math>\triangle{BED}</math> intersects the line <math>AB</math> in points <math>B</math> and <math>F \ne B</math>. What is <math>CF</math>?
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<math>\textbf{(A) } 28 \qquad \textbf{(B) } 20\sqrt{2} \qquad \textbf{(C) } 30 \qquad \textbf{(D) } 32 \qquad \textbf{(E) } 20\sqrt{3}</math>
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==Solution==
 
==Solution==
  
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- kevinmathz
 
- kevinmathz
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{{AMC12 box|year=2021 Fall|ab=B|num-a=25|num-b=23}}
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{{MAA Notice}}

Revision as of 17:55, 25 November 2021

Problem

Triangle $ABC$ has side lengths $AB = 11, BC=24$, and $CA = 20$. The bisector of $\angle{BAC}$ intersects $\overline{BC}$ in point $D$, and intersects the circumcircle of $\triangle{ABC}$ in point $E \ne A$. The circumcircle of $\triangle{BED}$ intersects the line $AB$ in points $B$ and $F \ne B$. What is $CF$?

$\textbf{(A) } 28 \qquad \textbf{(B) } 20\sqrt{2} \qquad \textbf{(C) } 30 \qquad \textbf{(D) } 32 \qquad \textbf{(E) } 20\sqrt{3}$

Solution

Claim: $\triangle ADC \sim \triangle ABE.$

Proof: Note that $\angle CAD = \angle CAE = \angle EAB$ and $\angle DCA = \angle BCA = \angle BEA$ meaning that our claim is true by AA similarity.

Because of this similarity, we have that \[\frac{AC}{AD} = \frac{AE}{AB} \to AB \cdot AC = AD \cdot AE = AB \cdot AF\] by Power of a Point. Thus, $AC=AF=20.$

Now, note that $\angle CAF = \angle CAB$ and plug into Law of Cosines to find the angle's cosine: \[AB^2+AC^2-2\cdot AB \cdot AC \cdot \cos(\angle CAB) = BC^2 \to \cos(\angle CAB) = -\frac{1}{8}.\]

So, we observe that we can use Law of Cosines again to find $CF$: \[AF^2+AC^2-2 \cdot AF \cdot AC \cdot \cos(\angle CAF) = CF^2 \to CF = \boxed{\textbf{(C) }30}.\]

- kevinmathz

2021 Fall AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 23 		Followed by
Problem 25
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All AMC 12 Problems and Solutions

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