Art of Problem Solving
Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

2025 AMC 12A Problems/Problem 8: Difference between revisions

← Older editNewer edit →
No edit summary
Line 34: Line 34:
== Solution 1==
== Solution 1==


We will scale down the diagram by a factor of <imath>3</imath> so that <imath>AB = 3</imath> and <imath>AD = 8</imath>. Because <imath>\angle BEC = 30^{\circ}</imath>, then <imath>\angle BAC = \angle BDC = 30^{\circ}</imath> because they all subtend the same arc. Similarly, because <imath>\angle CED = 30^{\circ}</imath>, <imath>\angle CAD = \angle CBD = 30^{\circ}</imath> as well.
We will scale down the diagram by a factor of <imath>3</imath> so that <imath>AB = 3</imath> and <imath>AD = 8.</imath> Because <imath>\angle BEC = 30^{\circ},</imath> then <imath>\angle BAC = \angle BDC = 30^{\circ}</imath> because they all subtend the same arc. Similarly, because <imath>\angle CED = 30^{\circ},</imath> <imath>\angle CAD = \angle CBD = 30^{\circ}</imath> as well.


Notice <imath>\triangle ABD</imath>, which has <imath>\angle BAD = 60</imath>. Applying Law of Cosines, we get:
Notice <imath>\triangle ABD,</imath> which has <imath>\angle BAD = 60.</imath> Applying Law of Cosines, we get:
<cmath>BD^2 = AB^2+AD^2-2AB\cdot AD \cdot\cos{60^{\circ}}</cmath>
<cmath>BD^2 = AB^2+AD^2-2AB\cdot AD \cdot\cos{60^{\circ}}</cmath>
<cmath>BD^2 = 9 + 64 - 2 \cdot 3 \cdot 8 \cdot \frac{1}{2}</cmath>
<cmath>BD^2 = 9 + 64 - 2 \cdot 3 \cdot 8 \cdot \frac{1}{2}</cmath>
<cmath>BD^2 = 49.</cmath>
<cmath>BD^2 = 49.</cmath>
So, <imath>BD = 7</imath>. From here, we want <imath>BF</imath>. Noticing that <imath>AF</imath> is the angle bisector of <imath>\angle BAD</imath>, we apply the Angle Bisector Theorem:
So, <imath>BD = 7.</imath> From here, we want <imath>BF.</imath> Noticing that <imath>AF</imath> is the angle bisector of <imath>\angle BAD,</imath> we apply the Angle Bisector Theorem:
<cmath>\frac{AB}{BF} = \frac{AD}{DF}</cmath>
<cmath>\frac{AB}{BF} = \frac{AD}{DF}</cmath>
<cmath>\frac{3}{BF}=\frac{8}{7-BF}.</cmath>
<cmath>\frac{3}{BF}=\frac{8}{7-BF}.</cmath>
Solving for <imath>BF</imath>, we get <imath>BF = \frac{21}{11}.</imath> Remember to scale the figure back up by a factor of <imath>3</imath>, so our answer is <imath>\frac{21}{11}\cdot 3 = \boxed{\frac{63}{11}}.</imath>
Solving for <imath>BF,</imath> we get <imath>BF = \frac{21}{11}.</imath> Remember to scale the figure back up by a factor of <imath>3,</imath> so our answer is <imath>\frac{21}{11}\cdot 3 = \boxed{\frac{63}{11}}.</imath>


~lprado
~lprado

Revision as of 11:42, 11 November 2025

Problem

Pentagon $ABCDE$ is inscribed in a circle, and $\angle BEC = \angle CED = 30^\circ$. Let line $AC$ and line $BD$ intersect at point $F$, and suppose that $AB = 9$ and $AD = 24$. What is $BF$?

$\textbf{(A) } \frac{57}{11} \qquad\textbf{(B) } \frac{59}{11} \qquad\textbf{(C) } \frac{60}{11} \qquad\textbf{(D) } \frac{61}{11} \qquad\textbf{(E) } \frac{63}{11}$

Diagram

[asy] /* Made by MRENTHUSIASM */ size(200); real r = 7*sqrt(3); pair O, A, B, C, D, E, F; O = origin; B = r*dir(30); C = r*dir(-30); D = r*dir(-90); E = r*dir(180); A = intersectionpoints(Circle(O,r),Circle(B,9))[0]; F = intersectionpoint(A--C,B--D); draw(Circle(O,r)^^A--B--C--D--E--cycle^^D--B--E--C--A--cycle); dot("$B$",B,1.5*B/r,linewidth(4)); dot("$C$",C,1.5*C/r,linewidth(4)); dot("$D$",D,1.5*D/r,linewidth(4)); dot("$E$",E,1.5*E/r,linewidth(4)); dot("$A$",A,1.5*A/r,linewidth(4)); dot("$F$",F,1.5*F/r,linewidth(4)); label("$30^{\circ}$",E,6*(1,0),fontsize(9)); label("$30^{\circ}$",E,7*dir(-32),fontsize(9)); label("$9$",0.9*midpoint(A--B)); label("$24$",1.8*midpoint(A--D)); [/asy] ~MRENTHUSIASM

Solution 1

We will scale down the diagram by a factor of $3$ so that $AB = 3$ and $AD = 8.$ Because $\angle BEC = 30^{\circ},$ then $\angle BAC = \angle BDC = 30^{\circ}$ because they all subtend the same arc. Similarly, because $\angle CED = 30^{\circ},$ $\angle CAD = \angle CBD = 30^{\circ}$ as well.

Notice $\triangle ABD,$ which has $\angle BAD = 60.$ Applying Law of Cosines, we get: \[BD^2 = AB^2+AD^2-2AB\cdot AD \cdot\cos{60^{\circ}}\] \[BD^2 = 9 + 64 - 2 \cdot 3 \cdot 8 \cdot \frac{1}{2}\] \[BD^2 = 49.\] So, $BD = 7.$ From here, we want $BF.$ Noticing that $AF$ is the angle bisector of $\angle BAD,$ we apply the Angle Bisector Theorem: \[\frac{AB}{BF} = \frac{AD}{DF}\] \[\frac{3}{BF}=\frac{8}{7-BF}.\] Solving for $BF,$ we get $BF = \frac{21}{11}.$ Remember to scale the figure back up by a factor of $3,$ so our answer is $\frac{21}{11}\cdot 3 = \boxed{\frac{63}{11}}.$

~lprado

Solution 2 Law of (Co)Sine

From cyclic quadrilateral $CDAE$, we have $\angle CAD = \angle CED = 30^\circ.$ Since $ABDE$ is also cyclic, we have $\angle BAD = \angle BED = 60^\circ$, so, \[\angle BAC= \angle BAD - \angle CAD = 60^\circ - 30^\circ = 30^\circ.\] Using Law of Cosines on $\triangle ABD$, we get \[BD^2=9^2+24^2-2(9)(24)\cos(60^\circ).\] Solving, we get $BD=21$. Next, let $\overline{BF}=x$, and $\angle AFB = \theta$, which means $\overline{FD}=21-x$ and $\angle AFD = 180-\theta$. Using Law of Sines on $\triangle AFB$, we have \[\frac{9}{\sin \theta}=\frac{x}{\sin 30}.\] Solving for $\sin \theta$, we get $\sin \theta = \frac{9}{2x}$. Now we apply the Law of Sines to $\triangle AFD.$ We have \[\frac{24}{\sin(180-\theta)} = \frac{21-x}{\sin 30}.\] Since $\sin(180-\theta) = \sin(\theta),$ and $\sin \theta = \frac{9}{2x}$, we have \[\frac{16x}{3} = 42-2x.\] Solving for $x$ gives $\boxed{x=\frac{63}{11}}$ or $\boxed{\text{E}}$.

~evanhliu2009

Solution 3 (Ptolemy’s + Similarity)

We have $ABCDE$ cyclic, so $\angle BAC=\angle CAD=\angle BEC=30^\circ$. Hence cyclic quadrilateral $ABCD$ has $\angle BAD=60^\circ$. Law of Cosines on triangle $BAD$ gives $\overline{BD}^2=9^2+24^2-2\cdot9\cdot24\cos60^\circ$. Hence $\overline{BD}=21$. Since triangle $BCD$ is a 120-30-30 triangle, we can use law of sines or just memorize ratios to get $\overline{BC}=\overline{CD}=7\sqrt3$. Now Ptolemy’s on $ABCD$ yields $7\sqrt3(9+24)=21\overline{AC}$. Hence $\overline{AC}=11\sqrt3$. Now notice that $\angle BCF=\angle ACB$, and $\angle CBF=\angle CAB=30^\circ$. Hence triangles $CBF$ and $CAB$ are similar, and $\frac{\overline{BF}}{\overline{BC}}=\frac{\overline{AB}}{\overline{AC}}$, so $\frac{\overline{BF}}{7\sqrt3}=\frac9{11\sqrt3}$ and $\overline{BF}=\frac{63}{11}$, or $\boxed{\textit{E}}$.

~benjamintontungtungtungsahur (look guys im famous)

Video Solution by Power Solve

https://youtu.be/Vd_kvodRjNQ?si=ZuoUjGLXcZter8PB&t=753

Video Solution 4 by SpreadTheMathLove

https://www.youtube.com/watch?v=ycwWI10M244

See Also

2025 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 7 		Followed by
Problem 9
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 12 Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America.

Retrieved from "https://wiki.aopstest.com/wiki/index.php?title=2025_AMC_12A_Problems/Problem_8&oldid=268578"