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

Difference between revisions of "1985 AHSME Problems/Problem 28"

m (Fixed punctuation)
(Fixed LaTeX in the second solution)
Line 52: Line 52:
  
 
== Solution 2 ==
 
== Solution 2 ==
−
Let angle A be equal to <math>x</math> degrees. Then angle C is equal to <math>3x</math> degrees, and angle B is equal to <math>180-4x</math> degrees. Let D be a point on side AB such that angle ACD is equal to <math>x</math> degrees. Because <math>2x+180-4x+CDB=180</math>, angle CDB is equal to <math>2x</math> degrees. We can now see that triangles CDB and CDA are both isosceles. CB=DB and AD=AC. From isosceles triangle CDB, we now know that BD = 27, and since AB = <math>c</math> = 48, we know that AD = 21. From isosceles triangle CDA, we now know that CD = 21. Applying Stewart's Theorem on triangle ABC gives us AC = 35, which is <math>\boxed{\text{B}}</math>.
+
Let angle <math>A</math> be equal to <math>x</math> degrees. Then angle <math>C</math> is equal to <math>3x</math> degrees, and angle <math>B</math> is equal to <math>180-4x</math> degrees. Let <math>D</math> be a point on side <math>AB</math> such that <math>\angle ACD</math> is equal to <math>x</math> degrees. Because <math>2x+180-4x+\angle CDB=180</math>, angle <math>CDB</math> is equal to <math>2x</math> degrees. We can now see that triangles <math>CDB</math> and <math>CDA</math> are both isosceles, with <math>CB=DB</math> and <math>AD=AC</math>. From isosceles triangle <math>CDB</math>, we now know that <math>BD = 27</math>, and since <math>AB = c = 48</math>, we know that <math>AD = 21</math>. From isosceles triangle <math>CDA</math>, we now know that <math>CD = 21</math>. Applying Stewart's Theorem on triangle <math>ABC</math> gives us <math>AC = 35</math>, which is <math>\boxed{\text{B}}</math>.
  
 
==See Also==
 
==See Also==
 
{{AHSME box|year=1985|num-b=27|num-a=29}}
 
{{AHSME box|year=1985|num-b=27|num-a=29}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 00:18, 3 April 2018

Problem

In $\triangle ABC$, we have $\angle C=3\angle A, a=27$ and $c=48$. What is $b$?

[asy] defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, B=(14,0), C=(10,6); draw(A--B--C--cycle); label("$A$", A, SW); label("$B$", B, SE); label("$C$", C, N); label("$a$", B--C, dir(B--C)*dir(-90)); label("$b$", A--C, dir(C--A)*dir(-90)); label("$c$", A--B, dir(A--B)*dir(-90));[/asy]

$\mathrm{(A)\ } 33 \qquad \mathrm{(B) \ }35 \qquad \mathrm{(C) \ } 37 \qquad \mathrm{(D) \ } 39 \qquad \mathrm{(E) \ }\text{not uniquely determined}$

Solution 1

From the Law of Sines, we have $\frac{\sin(A)}{a}=\frac{\sin(C)}{c}$, or $\frac{\sin(A)}{27}=\frac{\sin(3A)}{48}\implies 9\sin(3A)=16\sin(A)$.

We now need to find an identity relating $\sin(3A)$ and $\sin(A)$. We have


$\sin(3A)=\sin(2A+A)=\sin(2A)\cos(A)+\cos(2A)\sin(A)$


$=2\sin(A)\cos^2(A)+\sin(A)\cos^2(A)-\sin^3(A)=3\sin(A)\cos^2(A)-\sin^3(A)$


$=3\sin(A)(1-\sin^2(A))-\sin^3(A)=3\sin(A)-4\sin^3(A)$.


Thus we have $9(3\sin(A)-4\sin^3(A))=27\sin(A)-36\sin^3(A)=16\sin(A)$

$\implies 36\sin^3(A)=11\sin(A)$.

Therefore, $\sin(A)=0, \frac{\sqrt{11}}{6},$ or $-\frac{\sqrt{11}}{6}$. Notice that we must have $0^\circ<A<45^\circ$ because otherwise $A+3A>180^\circ$. We can therefore disregard $\sin(A)=0$ because then $A=0$ and also we can disregard $\sin(A)=-\frac{\sqrt{11}}{6}$ because then $A$ would be in the third or fourth quadrants, much greater than the desired range.


Therefore, $\sin(A)=\frac{\sqrt{11}}{6}$, and $\cos(A)=\sqrt{1-\left(\frac{\sqrt{11}}{6}\right)^2}=\frac{5}{6}$. Going back to the Law of Sines, we have $\frac{\sin(A)}{27}=\frac{\sin(B)}{b}=\frac{\sin(\pi-3A-A)}{b}=\frac{\sin(4A)}{b}$.


We now need to find $\sin(4A)$.

$\sin(4A)=\sin(2\cdot2A)=2\sin(2A)\cos(2A)$

$=2(2\sin(A)\cos(A))(\cos^2(A)-\sin^2(A))$

$=2\cdot2\cdot\frac{\sqrt{11}}{6}\cdot\frac{5}{6}\left(\left(\frac{5}{6}\right)^2-\left(\frac{\sqrt{11}}{6}\right)^2\right)=\frac{35\sqrt{11}}{162}$.


Therefore, $\frac{\frac{\sqrt{11}}{6}}{27}=\frac{\frac{35\sqrt{11}}{162}}{b}\implies b=\frac{6\cdot27\cdot35}{162}=35, \boxed{\text{B}}$.

Solution 2

Let angle $A$ be equal to $x$ degrees. Then angle $C$ is equal to $3x$ degrees, and angle $B$ is equal to $180-4x$ degrees. Let $D$ be a point on side $AB$ such that $\angle ACD$ is equal to $x$ degrees. Because $2x+180-4x+\angle CDB=180$, angle $CDB$ is equal to $2x$ degrees. We can now see that triangles $CDB$ and $CDA$ are both isosceles, with $CB=DB$ and $AD=AC$. From isosceles triangle $CDB$, we now know that $BD = 27$, and since $AB = c = 48$, we know that $AD = 21$. From isosceles triangle $CDA$, we now know that $CD = 21$. Applying Stewart's Theorem on triangle $ABC$ gives us $AC = 35$, which is $\boxed{\text{B}}$.

See Also

1985 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 27 		Followed by
Problem 29
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 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1985_AHSME_Problems/Problem_28&oldid=93796"