Difference between revisions of "2006 AMC 10B Problems/Problem 24"
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== Problem == | == Problem == | ||
− | + | [[Circle]]s with centers <math>O</math> and <math>P</math> have radii <math>2</math> and <math>4</math>, respectively, and are externally tangent. Points <math>A</math> and <math>B</math> on the circle with center <math>O</math> and points <math>C</math> and <math>D</math> on the circle with center <math>P</math> are such that <math>AD</math> and <math>BC</math> are common external tangents to the circles. What is the area of the [[concave]] [[hexagon]] <math>AOBCPD</math>? | |
− | + | <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> | ||
<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== |
− | Since a tangent line is perpendicular to the radius containing the | + | https://youtu.be/cdjZ9Xd3Yt8 |
+ | |||
+ | ~ 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 | ||
+ | |||
+ | 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, <math>\angle OAD = \angle PDA = 90^\circ</math>. | ||
− | Construct a perpendicular to <math>DP</math> that goes through point <math>O</math>. Label the point of intersection <math>X</math>. | + | Construct a perpendicular to <math>DP</math> that goes through point <math>O</math>. Label the point of [[intersection]] <math>X</math>. |
− | Clearly <math>OADX</math> is a rectangle. | + | Clearly <math>OADX</math> is a [[rectangle]], so <math>DX=2</math> and <math>PX=2</math>. By the [[Pythagorean Theorem]], <math>OX = \sqrt{6^2 - 2^2} = 4\sqrt{2}</math>. |
− | + | The area of <math>OADX</math> is <math>2\cdot4\sqrt{2}=8\sqrt{2}</math>. The area of <math>OXP</math> is <math>\frac{1}{2}\cdot2\cdot4\sqrt{2}=4\sqrt{2}</math>, so the area of quadrilateral <math>OADP</math> is <math>8\sqrt{2}+4\sqrt{2}=12\sqrt{2}</math>. Using similar steps, the area of quadrilateral <math>OBCP</math> is also <math>12\sqrt{2}</math>. Therefore, the area of hexagon <math>AOBCPD</math> is <math>2\cdot12\sqrt{2}= 24\sqrt{2} \Longrightarrow \boxed{\mathrm{(B)}}</math>. | |
− | |||
− | |||
− | + | Note: Quadrilaterals <math>OADP</math> and <math>OBCP</math> are congruent, so they have equal areas. | |
− | + | == Video Solution == | |
− | + | https://www.youtube.com/watch?v=EmUfNAxLZ7A ~David | |
− | + | == See also == | |
+ | {{AMC10 box|year=2006|ab=B|num-b=23|num-a=25}} | ||
− | + | [[Category:Introductory Geometry Problems]] | |
− | + | [[Category:Area Problems]] | |
− | + | [[Category:Circle Problems]] | |
+ | {{MAA Notice}} |
Latest revision as of 13:38, 6 April 2024
Contents
Problem
Circles with centers and have radii and , respectively, and are externally tangent. Points and on the circle with center and points and on the circle with center are such that and are common external tangents to the circles. What is the area of the concave hexagon ?
Video Solution 1
~ Education, the Study of Everything
Solution 1 (Similar Triangles)
When we see this problem, it practically screams similar triangles at us. Extend to the left until it intersects lines and at point . Triangles and are similar, and by symmetry, so are triangles and . Then, the area of kite is just and the area of kite is (using our similarity ratios). The difference of these yields the area of hexagon or .
~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, .
Construct a perpendicular to that goes through point . Label the point of intersection .
Clearly is a rectangle, so and . By the Pythagorean Theorem, .
The area of is . The area of is , so the area of quadrilateral is . Using similar steps, the area of quadrilateral is also . Therefore, the area of hexagon is .
Note: Quadrilaterals and are congruent, so they have equal areas.
Video Solution
https://www.youtube.com/watch?v=EmUfNAxLZ7A ~David
See also
2006 AMC 10B (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.