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 "2006 AMC 10B Problems/Problem 24"

m (→‎Solution)
(→‎See also)
 
(8 intermediate revisions by 6 users not shown)
Line 25: Line 25:
 
<math> \mathrm{(A) \ } 18\sqrt{3}\qquad \mathrm{(B) \ } 24\sqrt{2}\qquad \mathrm{(C) \ } 36\qquad \mathrm{(D) \ } 24\sqrt{3}\qquad \mathrm{(E) \ } 32\sqrt{2} </math>
 
<math> \mathrm{(A) \ } 18\sqrt{3}\qquad \mathrm{(B) \ } 24\sqrt{2}\qquad \mathrm{(C) \ } 36\qquad \mathrm{(D) \ } 24\sqrt{3}\qquad \mathrm{(E) \ } 32\sqrt{2} </math>
  
−
== Solution ==
+
==Video Solution 1==
 +
https://youtu.be/cdjZ9Xd3Yt8
 +
[Class:Geometry]
 +
~ Education, the Study of Everything
 +
 
 +
== Solution 1 (Similar Triangles) ==
 +
 
 +
When we see this problem, it practically screams similar triangles at us. Extend <math>OP</math> to the left until it intersects lines <math>AD</math> and <math>BC</math> at point <math>E</math>. Triangles <math>EBO</math> and <math>ECP</math> are similar, and by symmetry, so are triangles <math>EAO</math> and <math>EDP</math>. Then, the area of kite <math>EAOB</math> is just <math>4 \sqrt{2} \times 2</math> and the area of kite <math>EDPC</math> is <math>8 \sqrt{2} \times 4</math> (using our similarity ratios). The difference of these yields the area of hexagon <math>AOBCPD</math> or <math>24\sqrt{2} \Longrightarrow \boxed{\mathrm{(B)}}</math>.
 +
 
 +
~icecreamrolls8
 +
 
 +
== Solution 2 (Perpendiculars) ==
  
 
File is too big, so go to https://www.imgur.com/a/7aphGaa
 
File is too big, so go to https://www.imgur.com/a/7aphGaa
Line 42: Line 53:
 
Note: Quadrilaterals <math>OADP</math> and <math>OBCP</math> are congruent, so they have equal areas.
 
Note: Quadrilaterals <math>OADP</math> and <math>OBCP</math> are congruent, so they have equal areas.
  
−
== See also ==
+
== Video Solution ==
−
{{AMC10 box|year=2006|ab=B|num-b=23|num-a=25}}
+
 
 +
https://www.youtube.com/watch?v=EmUfNAxLZ7A    ~David
 +
 
  
−
[[Category:Introductory Geometry Problems]]
 
−
[[Category:Area Problems]]
 
−
[[Category:Circle Problems]]
 
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 13:41, 4 November 2024

Problem

Circles with centers $O$ and $P$ have radii $2$ and $4$, respectively, and are externally tangent. Points $A$ and $B$ on the circle with center $O$ and points $C$ and $D$ on the circle with center $P$ are such that $AD$ and $BC$ are common external tangents to the circles. What is the area of the concave hexagon $AOBCPD$?

[asy]size(200);defaultpen(linewidth(0.8)); pair X=(-6,0), O=origin, P=(6,0), B=tangent(X, O, 2, 1), A=tangent(X, O, 2, 2), C=tangent(X, P, 4, 1), D=tangent(X, P, 4, 2); pair top=X+15*dir(X--A), bottom=X+15*dir(X--B); draw(Circle(O, 2)^^Circle(P, 4)); draw(bottom--X--top); draw(A--O--B^^O--P^^D--P--C); pair point=X; label("$2$", midpoint(O--A), dir(point--midpoint(O--A))); label("$4$", midpoint(P--D), dir(point--midpoint(P--D))); label("$O$", O, SE); label("$P$", P, dir(point--P)); pair point=O; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); pair point=P; label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); fill((-3,7)--(-3,-7)--(-7,-7)--(-7,7)--cycle, white);[/asy]

$\mathrm{(A) \ } 18\sqrt{3}\qquad \mathrm{(B) \ } 24\sqrt{2}\qquad \mathrm{(C) \ } 36\qquad \mathrm{(D) \ } 24\sqrt{3}\qquad \mathrm{(E) \ } 32\sqrt{2}$

Video Solution 1

https://youtu.be/cdjZ9Xd3Yt8 [Class:Geometry] ~ Education, the Study of Everything

Solution 1 (Similar Triangles)

When we see this problem, it practically screams similar triangles at us. Extend $OP$ to the left until it intersects lines $AD$ and $BC$ at point $E$. Triangles $EBO$ and $ECP$ are similar, and by symmetry, so are triangles $EAO$ and $EDP$. Then, the area of kite $EAOB$ is just $4 \sqrt{2} \times 2$ and the area of kite $EDPC$ is $8 \sqrt{2} \times 4$ (using our similarity ratios). The difference of these yields the area of hexagon $AOBCPD$ or $24\sqrt{2} \Longrightarrow \boxed{\mathrm{(B)}}$.

~icecreamrolls8

Solution 2 (Perpendiculars)

File is too big, so go to https://www.imgur.com/a/7aphGaa

Sorry for the wrong point names, I didn't know how to change them.

Since a tangent line is perpendicular to the radius containing the point of tangency, $\angle OAD = \angle PDA = 90^\circ$.

Construct a perpendicular to $DP$ that goes through point $O$. Label the point of intersection $X$.

Clearly $OADX$ is a rectangle, so $DX=2$ and $PX=2$. By the Pythagorean Theorem, $OX = \sqrt{6^2 - 2^2} = 4\sqrt{2}$.

The area of $OADX$ is $2\cdot4\sqrt{2}=8\sqrt{2}$. The area of $OXP$ is $\frac{1}{2}\cdot2\cdot4\sqrt{2}=4\sqrt{2}$, so the area of quadrilateral $OADP$ is $8\sqrt{2}+4\sqrt{2}=12\sqrt{2}$. Using similar steps, the area of quadrilateral $OBCP$ is also $12\sqrt{2}$. Therefore, the area of hexagon $AOBCPD$ is $2\cdot12\sqrt{2}= 24\sqrt{2} \Longrightarrow \boxed{\mathrm{(B)}}$.


Note: Quadrilaterals $OADP$ and $OBCP$ are congruent, so they have equal areas.

Video Solution

https://www.youtube.com/watch?v=EmUfNAxLZ7A ~David


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=2006_AMC_10B_Problems/Problem_24&oldid=231555"