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 "2007 AMC 10A Problems/Problem 24"

(→‎Solution)
(→‎Solution)
 
(12 intermediate revisions by 9 users not shown)
Line 2: Line 2:
 
[[Circle]]s centered at <math>A</math> and <math>B</math> each have radius <math>2</math>, as shown. Point <math>O</math> is the [[midpoint]] of <math>\overline{AB}</math>, and <math>OA = 2\sqrt {2}</math>. Segments <math>OC</math> and <math>OD</math> are tangent to the circles centered at <math>A</math> and <math>B</math>, respectively, and <math>EF</math> is a common [[tangent (geometry)|tangent]]. What is the area of the shaded region <math>ECODF</math>?
 
[[Circle]]s centered at <math>A</math> and <math>B</math> each have radius <math>2</math>, as shown. Point <math>O</math> is the [[midpoint]] of <math>\overline{AB}</math>, and <math>OA = 2\sqrt {2}</math>. Segments <math>OC</math> and <math>OD</math> are tangent to the circles centered at <math>A</math> and <math>B</math>, respectively, and <math>EF</math> is a common [[tangent (geometry)|tangent]]. What is the area of the shaded region <math>ECODF</math>?
  
−
[[Image:2007 AMC 10A problem 24.png]]
+
<asy>
 +
size(5cm);
 +
pair A=(-2*sqrt(2),0), B = (2*sqrt(2),0), C = A+2*dir(45), D = B+2*dir(135), E = A+2*dir(90), F = B+2*dir(90);
 +
 
 +
fill((0,0)--C--E--F--D--cycle,gray(0.6));
 +
unfill(circle(A,2));
 +
unfill(circle(B,2));
 +
draw(circle(A,2));
 +
draw(circle(B,2));
 +
draw(E--F);
 +
draw(C--(0,0)--D);
 +
draw(A+2*dir(300)--A--B--B+2*dir(240));
 +
 
 +
dot((0,0));
 +
dot(A);
 +
dot(B);
 +
dot(C);
 +
dot(D);
 +
dot(E);
 +
dot(F);
 +
 
 +
label("$A$",A,dir(180));
 +
label("$B$",B,dir(0));
 +
label("$C$",C,dir(240));
 +
label("$D$",D,dir(300));
 +
label("$E$",E,dir(90));
 +
label("$F$",F,dir(90));
 +
label("$O$",(0,0),dir(270));
 +
label("$2$",A+dir(300),dir(210));
 +
label("$2$",B+dir(240),dir(330));
 +
</asy>
  
 
<math>\text{(A)}\ \frac {8\sqrt {2}}{3} \qquad \text{(B)}\ 8\sqrt {2} - 4 - \pi \qquad \text{(C)}\ 4\sqrt {2} \qquad \text{(D)}\ 4\sqrt {2} + \frac {\pi}{8} \qquad \text{(E)}\ 8\sqrt {2} - 2 - \frac {\pi}{2}</math>
 
<math>\text{(A)}\ \frac {8\sqrt {2}}{3} \qquad \text{(B)}\ 8\sqrt {2} - 4 - \pi \qquad \text{(C)}\ 4\sqrt {2} \qquad \text{(D)}\ 4\sqrt {2} + \frac {\pi}{8} \qquad \text{(E)}\ 8\sqrt {2} - 2 - \frac {\pi}{2}</math>
  
 
==Solution==
 
==Solution==
−
The area we are trying to find is simply <math>ABFE-(\arc{AEC}+\triangle{ACO}+\triangle{BDO}+\arc{BFD})</math>.
+
The area we are trying to find is simply <math>ABFE-(\overarc{AEC}+\triangle{ACO}+\triangle{BDO}+\overarc{BFD}).\overline{EF}\parallel\overline{AB}</math>.  Thus, <math>ABFE</math> is a [[rectangle]], and so its area is <math>b\times{h}=2\times{(AO+OB)}=2\times{2(2\sqrt{2})}=8\sqrt{2}</math>.
−
Obviously, <math>\overline{EF}</math> is parallel to <math>\overline{AB}</math>.  Thus, <math>ABFE</math> is a [[rectangle]], and so its area is <math>b\times{h}=2\times{(AO+OB)}=2\times{2(2\sqrt{2})}=8\sqrt{2}</math>.
 
  
−
Since <math>\overline{OC}</math> is tangent to circle <math>A</math>, <math>\triangle{ACO}</math> is a right <math>\triangle</math>.  We know <math>AO=2\sqrt{2}</math> and <math>AC=2</math>, so <math>\triangle{ACO}</math> is isosceles, a <math>45</math>-<math>45</math> right <math>\triangle</math>, and has <math>\overline{CO}</math> with length <math>2</math>.  The area of <math>\triangle{ACO}=\frac{1}{2}bh=2</math>.  For obvious reasons, <math>\triangle{ACO}=\triangle{BDO}</math>, and so the area of <math>\triangle{BDO}</math> is also <math>2</math>.
+
Since <math>\overline{OC}</math> is tangent to circle <math>A</math>, <math>\triangle{ACO}</math> is a right triangle.  We know <math>AO=2\sqrt{2}</math> and <math>AC=2</math>, so <math>\triangle{ACO}</math> is an isosceles right triangle, and has <math>\overline{CO}</math> with length <math>2</math>.  The area of <math>\triangle{ACO}=\frac{1}{2}bh=2</math>.  By symmetry, <math>\triangle{ACO}\cong\triangle{BDO}</math>, and so the area of <math>\triangle{BDO}</math> is also <math>2</math>.
  
−
<math>\arc{AEC}</math> (or <math>\arc{BFD}</math>, for that matter) is <math>\frac{1}{8}</math> the area of its circle.  Thus <math>\arc{AEC}</math> and <math>\arc{BFD}</math> both have an area of <math>\frac{\pi}{2}</math>.
+
<math>\overarc{AEC}</math> (or <math>\overarc{BFD}</math>, for that matter) is <math>\frac{1}{8}</math> the area of its circle since <math>\angle{OAC}</math> is 45 degrees and <math>\angle{OAE}</math> forms a right [[triangle]].  Thus <math>\overarc{AEC}</math> and <math>\overarc{BFD}</math> both have an area of <math>\frac{\pi}{2}</math>.
  
 
Plugging all of these areas back into the original equation yields <math>8\sqrt{2}-(\frac{\pi}{2}+2+2+\frac{\pi}{2})=8\sqrt{2}-(4+\pi)=\boxed{8\sqrt{2}-4-\pi}\ \mathrm{(B)}</math>.
 
Plugging all of these areas back into the original equation yields <math>8\sqrt{2}-(\frac{\pi}{2}+2+2+\frac{\pi}{2})=8\sqrt{2}-(4+\pi)=\boxed{8\sqrt{2}-4-\pi}\ \mathrm{(B)}</math>.
Line 20: Line 49:
  
 
[[Category:Introductory Geometry Problems]]
 
[[Category:Introductory Geometry Problems]]
 +
{{MAA Notice}}

Latest revision as of 11:53, 6 August 2024

Problem

Circles centered at $A$ and $B$ each have radius $2$, as shown. Point $O$ is the midpoint of $\overline{AB}$, and $OA = 2\sqrt {2}$. Segments $OC$ and $OD$ are tangent to the circles centered at $A$ and $B$, respectively, and $EF$ is a common tangent. What is the area of the shaded region $ECODF$?

[asy] size(5cm); pair A=(-2*sqrt(2),0), B = (2*sqrt(2),0), C = A+2*dir(45), D = B+2*dir(135), E = A+2*dir(90), F = B+2*dir(90); fill((0,0)--C--E--F--D--cycle,gray(0.6)); unfill(circle(A,2)); unfill(circle(B,2)); draw(circle(A,2)); draw(circle(B,2)); draw(E--F); draw(C--(0,0)--D); draw(A+2*dir(300)--A--B--B+2*dir(240)); dot((0,0)); dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); label("$A$",A,dir(180)); label("$B$",B,dir(0)); label("$C$",C,dir(240)); label("$D$",D,dir(300)); label("$E$",E,dir(90)); label("$F$",F,dir(90)); label("$O$",(0,0),dir(270)); label("$2$",A+dir(300),dir(210)); label("$2$",B+dir(240),dir(330)); [/asy]

$\text{(A)}\ \frac {8\sqrt {2}}{3} \qquad \text{(B)}\ 8\sqrt {2} - 4 - \pi \qquad \text{(C)}\ 4\sqrt {2} \qquad \text{(D)}\ 4\sqrt {2} + \frac {\pi}{8} \qquad \text{(E)}\ 8\sqrt {2} - 2 - \frac {\pi}{2}$

Solution

The area we are trying to find is simply $ABFE-(\overarc{AEC}+\triangle{ACO}+\triangle{BDO}+\overarc{BFD}).\overline{EF}\parallel\overline{AB}$. Thus, $ABFE$ is a rectangle, and so its area is $b\times{h}=2\times{(AO+OB)}=2\times{2(2\sqrt{2})}=8\sqrt{2}$.

Since $\overline{OC}$ is tangent to circle $A$, $\triangle{ACO}$ is a right triangle. We know $AO=2\sqrt{2}$ and $AC=2$, so $\triangle{ACO}$ is an isosceles right triangle, and has $\overline{CO}$ with length $2$. The area of $\triangle{ACO}=\frac{1}{2}bh=2$. By symmetry, $\triangle{ACO}\cong\triangle{BDO}$, and so the area of $\triangle{BDO}$ is also $2$.

$\overarc{AEC}$ (or $\overarc{BFD}$, for that matter) is $\frac{1}{8}$ the area of its circle since $\angle{OAC}$ is 45 degrees and $\angle{OAE}$ forms a right triangle. Thus $\overarc{AEC}$ and $\overarc{BFD}$ both have an area of $\frac{\pi}{2}$.

Plugging all of these areas back into the original equation yields $8\sqrt{2}-(\frac{\pi}{2}+2+2+\frac{\pi}{2})=8\sqrt{2}-(4+\pi)=\boxed{8\sqrt{2}-4-\pi}\ \mathrm{(B)}$.

See also

2007 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 23 		Followed by
Problem 25
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 10 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=2007_AMC_10A_Problems/Problem_24&oldid=224574"