Difference between revisions of "2008 AMC 12B Problems/Problem 9"

Line 4: Line 4:
 
<math>\textbf{(A)}\ \sqrt {10} \qquad \textbf{(B)}\ \frac {7}{2} \qquad \textbf{(C)}\ \sqrt {14} \qquad \textbf{(D)}\ \sqrt {15} \qquad \textbf{(E)}\ 4</math>
 
<math>\textbf{(A)}\ \sqrt {10} \qquad \textbf{(B)}\ \frac {7}{2} \qquad \textbf{(C)}\ \sqrt {14} \qquad \textbf{(D)}\ \sqrt {15} \qquad \textbf{(E)}\ 4</math>
  
==Solution==
+
==Solutions==
===Trig Solution:===
+
===Solution 1===
Let <math>\alpha</math> be the angle that subtends the arc AB. By the law of cosines,
+
Let <math>\alpha</math> be the angle that subtends the arc <math>AB</math>. By the law of cosines,
<math>6^2=5^2+5^2-2*5*5cos(\alpha)</math>
+
<math>6^2=5^2+5^2-2\cdot 5\cdot 5\cos(\alpha)</math> implies <math>\cos(\alpha) = 7/25</math>.
  
<math>\alpha = cos^{-1}(7/25)</math>
+
The [[Trigonometric_identities#Half_Angle_Identities | half-angle formula]] says that
 +
<math>\cos(\alpha/2) = \frac{\sqrt{1+\cos(\alpha)}}{2} = \sqrt{\frac{32/25}{2}} = \sqrt{\frac{16}{25}} = \frac{4}{5}</math>. The law of cosines tells us <math>AC = \sqrt{5^2+5^2-2*5\cdot 5\cdot \frac{4}{5}} = \sqrt{50-50\frac{4}{5}} = \sqrt{10}</math>, which is answer choice <math>\boxed{\text{A}}</math>.
  
The half-angle formula says that
+
===Solution 2===
<math>cos(\alpha/2) = \frac{\sqrt{1+cos(\alpha)}}{2} = \sqrt{\frac{32/25}{2}} = \sqrt{\frac{16}{25}} = \frac{4}{5}</math>
+
{{asy image|
<math>AC = \sqrt{5^2+5^2-2*5*5*\frac{4}{5}}</math>
+
<asy>
 
+
defaultpen(fontsize(8));
<math>AC = \sqrt{50-50\frac{4}{5}}</math>
+
pair A=(-3,4), B=(3,4), C=(0,5), D=(0,4), O=(0,0);
 
+
D(Circle(O,5));
<math>AC = \sqrt{10}</math>, which is answer choice A.
+
D(O--B--A--O--C);D(A--C--B);
 
+
label("$A$",A,(-1,1));label("$O$",O,(0,-1));label("$B$",B,(1,1));label("$C$",C,(0,1));label("$D$",D,(-1,-1));
===More Elegant Solution===
+
</asy>
Define D as the midpoint of AB, and R the center of the circle. R, C, and D are collinear, and since D is the midpoint of AB,  
+
|right|Figure 1
 
+
}}
<math>m\angle RDA=90\deg</math>
+
Define <math>D</math> as the midpoint of line segment <math>\overline{AB}</math>, and <math>O</math> the center of the circle. Then <math>O</math>, <math>C</math>, and <math>D</math> are collinear, and since <math>D</math> is the midpoint of <math>AB</math>, <math>m\angle ODA=90\deg</math> and so <math>OD=\sqrt{5^2-3^2}=4</math>. Since <math>OD=4</math>, <math>CD=5-4=1</math>, and so <math>AC=\sqrt{3^2+1^2}=\sqrt{10} \rightarrow \boxed{\text{A}}</math>.
 
 
and so  
 
 
 
<math>RD=\sqrt{5^2-3^2}=4</math>.  
 
 
 
Since  
 
 
 
<math>RD=4</math>,
 
 
 
<math>CD=5-4=1</math>,
 
 
 
and so  
 
 
 
<math>AC=\sqrt{3^2+1^2}=\sqrt{10} \rightarrow A</math>
 
  
 
==See Also==
 
==See Also==
 
{{AMC12 box|year=2008|ab=B|num-b=8|num-a=10}}
 
{{AMC12 box|year=2008|ab=B|num-b=8|num-a=10}}

Revision as of 00:21, 15 August 2011

Problem 9

Points $A$ and $B$ are on a circle of radius $5$ and $AB = 6$. Point $C$ is the midpoint of the minor arc $AB$. What is the length of the line segment $AC$?

$\textbf{(A)}\ \sqrt {10} \qquad \textbf{(B)}\ \frac {7}{2} \qquad \textbf{(C)}\ \sqrt {14} \qquad \textbf{(D)}\ \sqrt {15} \qquad \textbf{(E)}\ 4$

Solutions

Solution 1

Let $\alpha$ be the angle that subtends the arc $AB$. By the law of cosines, $6^2=5^2+5^2-2\cdot 5\cdot 5\cos(\alpha)$ implies $\cos(\alpha) = 7/25$.

The half-angle formula says that $\cos(\alpha/2) = \frac{\sqrt{1+\cos(\alpha)}}{2} = \sqrt{\frac{32/25}{2}} = \sqrt{\frac{16}{25}} = \frac{4}{5}$. The law of cosines tells us $AC = \sqrt{5^2+5^2-2*5\cdot 5\cdot \frac{4}{5}} = \sqrt{50-50\frac{4}{5}} = \sqrt{10}$, which is answer choice $\boxed{\text{A}}$.

Solution 2

[asy] defaultpen(fontsize(8)); pair A=(-3,4), B=(3,4), C=(0,5), D=(0,4), O=(0,0); D(Circle(O,5)); D(O--B--A--O--C);D(A--C--B); label("$A$",A,(-1,1));label("$O$",O,(0,-1));label("$B$",B,(1,1));label("$C$",C,(0,1));label("$D$",D,(-1,-1)); [/asy]

Enlarge.png
Figure 1

Define $D$ as the midpoint of line segment $\overline{AB}$, and $O$ the center of the circle. Then $O$, $C$, and $D$ are collinear, and since $D$ is the midpoint of $AB$, $m\angle ODA=90\deg$ and so $OD=\sqrt{5^2-3^2}=4$. Since $OD=4$, $CD=5-4=1$, and so $AC=\sqrt{3^2+1^2}=\sqrt{10} \rightarrow \boxed{\text{A}}$.

See Also

2008 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 8
Followed by
Problem 10
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
Invalid username
Login to AoPS