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Difference between revisions of "1985 AIME Problems/Problem 9"
Line 58: Line 58:
 
MP("3",(A2+A3)/2,E);
 
MP("3",(A2+A3)/2,E);
 
MP("4",(A1+A3)/2,E);
 
MP("4",(A1+A3)/2,E);
−
D(anglemark(A2,O,A1,5)); D(anglemark(A3,O,A2,5));
+
D(anglemark(A2,O,A1,5)); D(anglemark(A3,O,A2,5)); D(anglemark(A2,A3,A1,10));
 
label("\(\alpha\)",(0.07,0.16),NE,fontsize(8));
 
label("\(\alpha\)",(0.07,0.16),NE,fontsize(8));
 
label("\(\beta\)",(0.12,-0.16),NE,fontsize(8));
 
label("\(\beta\)",(0.12,-0.16),NE,fontsize(8));
−
label("\(\alpha\)/2",(0.6,0.16),NE,fontsize(8));
+
label("\(\alpha\)/2",(0.82,-1.25),NE,fontsize(8));
 
</asy></center>
 
</asy></center>
  
 +
It’s easy to see that the angle opposite the side 2 is <math>\frac{\alpha}{2}</math>, and using the [[Law of Cosines]], we get:  <cmath>2^2 = 3^2 + 4^2 - 2\cdot3\cdot4\cos\frac{\alpha}{2}</cmath> Which, rearranges to: <cmath>21 = 24cos\frac{\alpha}{2}</cmath> And, that gets that <cmath>cos\frac{\alpha}{2} = 7/8</cmath> and using that <math>\cos 2\theta = 2\cos^2 \theta - 1  we get that
 +
</math>\cos\alpha = 17/32<math>,
 +
which gives an answer of </math>\boxed{049}<math>
 +
 +
— Alexlikemath
  
 
==Solution 3 (trig)==  
 
==Solution 3 (trig)==  
Line 75: Line 80:
 
so by the composite sine identity
 
so by the composite sine identity
 
<cmath>\frac{2}{r}=\frac{1}{r}\sqrt{1-\frac{2.25}{r^2}}+\frac{1.5}{r}\sqrt{1-\frac{1}{r^2}}</cmath>
 
<cmath>\frac{2}{r}=\frac{1}{r}\sqrt{1-\frac{2.25}{r^2}}+\frac{1.5}{r}\sqrt{1-\frac{1}{r^2}}</cmath>
−
multiply both sides by <math>2r</math>, then subtract <math>\sqrt{4-\frac{9}{r^2}}</math> from both sides
+
multiply both sides by </math>2r<math>, then subtract </math>\sqrt{4-\frac{9}{r^2}}<math> from both sides
 
squaring both sides, we get
 
squaring both sides, we get
 
<cmath>16 - 8\sqrt{4-\frac{9}{r^2}} + 4 - \frac{9}{r^2}=9 - \frac{9}{r^2}</cmath>
 
<cmath>16 - 8\sqrt{4-\frac{9}{r^2}} + 4 - \frac{9}{r^2}=9 - \frac{9}{r^2}</cmath>
Line 84: Line 89:
 
so
 
so
 
<cmath>\cos(\alpha)=2(\frac{49}{64})-1=\frac{34}{64}=\frac{17}{32}</cmath>
 
<cmath>\cos(\alpha)=2(\frac{49}{64})-1=\frac{34}{64}=\frac{17}{32}</cmath>
−
and the answer is <math>17+32=\boxed{049}</math>
+
and the answer is </math>17+32=\boxed{049}$
  
 
== See also ==
 
== See also ==

Revision as of 23:07, 30 April 2019

Problem

In a circle, parallel chords of lengths 2, 3, and 4 determine central angles of $\alpha$, $\beta$, and $\alpha + \beta$ radians, respectively, where $\alpha + \beta < \pi$. If $\cos \alpha$, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator?

Solution 1

[asy] size(200); pointpen = black; pathpen = black + linewidth(0.8); real r = 8/15^0.5, a = 57.91, b = 93.135; pair O = (0,0), A = r*expi(pi/3); D(CR(O,r)); D(O--rotate(a/2)*A--rotate(-a/2)*A--cycle); D(O--rotate(b/2)*A--rotate(-b/2)*A--cycle); D(O--rotate((a+b)/2)*A--rotate(-(a+b)/2)*A--cycle); MP("2",(rotate(a/2)*A+rotate(-a/2)*A)/2,NE); MP("3",(rotate(b/2)*A+rotate(-b/2)*A)/2,NE); MP("4",(rotate((a+b)/2)*A+rotate(-(a+b)/2)*A)/2,NE); D(anglemark(rotate(-(a+b)/2)*A,O,rotate((a+b)/2)*A,5)); label("\(\alpha+\beta\)",(0.08,0.08),NE,fontsize(8)); [/asy]

All chords of a given length in a given circle subtend the same arc and therefore the same central angle. Thus, by the given, we can re-arrange our chords into a triangle with the circle as its circumcircle.

[asy] size(200); pointpen = black; pathpen = black + linewidth(0.8); real r = 8/15^0.5, a = 57.91, b = 93.135; pair O = (0,0), A = r*expi(pi/3), A1 = rotate(a/2)*A, A2 = rotate(-a/2)*A, A3 = rotate(-a/2-b)*A; D(CR(O,r)); D(O--A1--A2--cycle); D(O--A2--A3--cycle); D(O--A1--A3--cycle); MP("2",(A1+A2)/2,NE); MP("3",(A2+A3)/2,E); MP("4",(A1+A3)/2,E); D(anglemark(A2,O,A1,5)); D(anglemark(A3,O,A2,5)); label("\(\alpha\)",(0.07,0.16),NE,fontsize(8)); label("\(\beta\)",(0.12,-0.16),NE,fontsize(8)); [/asy]

This triangle has semiperimeter $\frac{2 + 3 + 4}{2}$ so by Heron's formula it has area $K = \sqrt{\frac92 \cdot \frac52 \cdot \frac32 \cdot \frac12} = \frac{3}{4}\sqrt{15}$. The area of a given triangle with sides of length $a, b, c$ and circumradius of length $R$ is also given by the formula $K = \frac{abc}{4R}$, so $\frac6R = \frac{3}{4}\sqrt{15}$ and $R = \frac8{\sqrt{15}}$.

Now, consider the triangle formed by two radii and the chord of length 2. This isosceles triangle has vertex angle $\alpha$, so by the Law of Cosines,

\[2^2 = R^2 + R^2 - 2R^2\cos \alpha \Longrightarrow \cos \alpha = \frac{2R^2 - 4}{2R^2} = \frac{17}{32}\] and the answer is $17 + 32 = \boxed{049}$.


Solution 2 (Law of cosines)

[asy] size(200); pointpen = black; pathpen = black + linewidth(0.8); real r = 8/15^0.5, a = 57.91, b = 93.135; pair O = (0,0), A = r*expi(pi/3), A1 = rotate(a/2)*A, A2 = rotate(-a/2)*A, A3 = rotate(-a/2-b)*A; D(CR(O,r)); D(O--A1--A2--cycle); D(O--A2--A3--cycle); D(O--A1--A3--cycle); MP("2",(A1+A2)/2,NE); MP("3",(A2+A3)/2,E); MP("4",(A1+A3)/2,E); D(anglemark(A2,O,A1,5)); D(anglemark(A3,O,A2,5)); D(anglemark(A2,A3,A1,10)); label("\(\alpha\)",(0.07,0.16),NE,fontsize(8)); label("\(\beta\)",(0.12,-0.16),NE,fontsize(8)); label("\(\alpha\)/2",(0.82,-1.25),NE,fontsize(8)); [/asy]

It’s easy to see that the angle opposite the side 2 is $\frac{\alpha}{2}$, and using the Law of Cosines, we get: \[2^2 = 3^2 + 4^2 - 2\cdot3\cdot4\cos\frac{\alpha}{2}\] Which, rearranges to: \[21 = 24cos\frac{\alpha}{2}\] And, that gets that \[cos\frac{\alpha}{2} = 7/8\] and using that $\cos 2\theta = 2\cos^2 \theta - 1 we get that$\cos\alpha = 17/32$, which gives an answer of$\boxed{049}$— Alexlikemath

==Solution 3 (trig)== Using the first diagram above, <cmath>\sin \frac{\alpha}{2} = \frac{1}{r}</cmath> <cmath>\sin \frac{\beta}{2} = \frac{1.5}{r}</cmath> <cmath>\sin(\frac{\alpha}{2}+\frac{\beta}{2})=\frac{2}{r}</cmath> by the Pythagorean trig identities, <cmath>\cos\frac{\alpha}{2}=\sqrt{1-\frac{1}{r^2}}</cmath> <cmath>\cos\frac{\beta}{2}=\sqrt{1-\frac{2.25}{r^2}}</cmath> so by the composite sine identity <cmath>\frac{2}{r}=\frac{1}{r}\sqrt{1-\frac{2.25}{r^2}}+\frac{1.5}{r}\sqrt{1-\frac{1}{r^2}}</cmath> multiply both sides by$ (Error compiling LaTeX. Unknown error_msg)2r$, then subtract$\sqrt{4-\frac{9}{r^2}}$from both sides squaring both sides, we get <cmath>16 - 8\sqrt{4-\frac{9}{r^2}} + 4 - \frac{9}{r^2}=9 - \frac{9}{r^2}</cmath> <cmath>\Longrightarrow 16+4=9+8\sqrt{4-\frac{9}{r^2}}\Longrightarrow\frac{11}{8}=\sqrt{4-\frac{9}{r^2}}\Longrightarrow\frac{121}{64}=4-\frac{9}{r^2}</cmath> <cmath>\Longrightarrow\frac{(256-121)r^2}{64}=9\Longrightarrow r^2= \frac{64}{15}</cmath> plugging this back in, <cmath>\cos^2(\frac{\alpha}{2})=1-\frac{15}{64}=\frac{49}{64}</cmath> so <cmath>\cos(\alpha)=2(\frac{49}{64})-1=\frac{34}{64}=\frac{17}{32}</cmath> and the answer is$17+32=\boxed{049}$

See also

1985 AIME (Problems • Answer Key • Resources)
Preceded by
Problem 8 		Followed by
Problem 10
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions
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