Difference between revisions of "1992 AHSME Problems/Problem 26"

(Created page with "== Problem == <asy> fill((1,0)--arc((1,0),2,180,225)--cycle,grey); fill((-1,0)--arc((-1,0),2,315,360)--cycle,grey); fill((0,-1)--arc((0,-1),2-sqrt(2),225,315)--cycle,grey); fill(...")
 
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</asy>
 
</asy>
  
Semicircle <math>AB</math> has center <math>C</math> and radius <math>1</math>. Point <math>D</math> is on <math>AB</math> and <math>\overline{CD}\orthogonal\overline{AB}</math>. Extend <math>\overline{BD}</math> and <math>\overline{AD}</math> to <math>E</math> and <math>F</math>, respectively, so that circular arcs <math>AE</math> and <math>BF</math> have <math>B</math> and <math>A</math> as their respective centers. Circular arc <math>EF</math> has center <math>D</math>. The area of the shaded "smile" <math>AEFBDA</math>, is
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Semicircle <math>\widehat{AB}</math> has center <math>C</math> and radius <math>1</math>. Point <math>D</math> is on <math>\widehat{AB}</math> and <math>\overline{CD}\perp\overline{AB}</math>. Extend <math>\overline{BD}</math> and <math>\overline{AD}</math> to <math>E</math> and <math>F</math>, respectively, so that circular arcs <math>\widehat{AE}</math> and <math>\widehat{BF}</math> have <math>B</math> and <math>A</math> as their respective centers. Circular arc <math>\widehat{EF}</math> has center <math>D</math>. The area of the shaded "smile" <math>AEFBDA</math>, is
  
<math>\text{(A) } \quad
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<math>\text{(A) } (2-\sqrt{2})\pi\quad
\text{(B) } \quad
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\text{(B) } 2\pi-\pi \sqrt{2}-1\quad
\text{(C) } \quad
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\text{(C) } (1-\frac{\sqrt{2}}{2})\pi\quad\\
\text{(D) } \quad
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\text{(D) } \frac{5\pi}{2}-\pi\sqrt{2}-1\quad
\text{(E) } </math>
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\text{(E) } (3-2\sqrt{2})\pi</math>
  
 
== Solution ==
 
== Solution ==
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{{AHSME box|year=1993|num-b=1|num-a=2}}   
 
{{AHSME box|year=1993|num-b=1|num-a=2}}   
  
[[Category:Introductory Algebra Problems]]
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[[Category: Intermediate Geometry Problems]]
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 21:32, 27 September 2014

Problem

[asy] fill((1,0)--arc((1,0),2,180,225)--cycle,grey); fill((-1,0)--arc((-1,0),2,315,360)--cycle,grey); fill((0,-1)--arc((0,-1),2-sqrt(2),225,315)--cycle,grey); fill((0,0)--arc((0,0),1,180,360)--cycle,white); draw((1,0)--arc((1,0),2,180,225)--(1,0),black+linewidth(1)); draw((-1,0)--arc((-1,0),2,315,360)--(-1,0),black+linewidth(1)); draw((0,0)--arc((0,0),1,180,360)--(0,0),black+linewidth(1)); draw(arc((0,-1),2-sqrt(2),225,315),black+linewidth(1)); draw((0,0)--(0,-1),black+linewidth(1)); MP("C",(0,0),N);MP("A",(-1,0),N);MP("B",(1,0),N); MP("D",(0,-.8),NW);MP("E",(1-sqrt(2),-sqrt(2)),SW);MP("F",(-1+sqrt(2),-sqrt(2)),SE); [/asy]

Semicircle $\widehat{AB}$ has center $C$ and radius $1$. Point $D$ is on $\widehat{AB}$ and $\overline{CD}\perp\overline{AB}$. Extend $\overline{BD}$ and $\overline{AD}$ to $E$ and $F$, respectively, so that circular arcs $\widehat{AE}$ and $\widehat{BF}$ have $B$ and $A$ as their respective centers. Circular arc $\widehat{EF}$ has center $D$. The area of the shaded "smile" $AEFBDA$, is

$\text{(A) } (2-\sqrt{2})\pi\quad \text{(B) } 2\pi-\pi \sqrt{2}-1\quad \text{(C) } (1-\frac{\sqrt{2}}{2})\pi\quad\\ \text{(D) } \frac{5\pi}{2}-\pi\sqrt{2}-1\quad \text{(E) } (3-2\sqrt{2})\pi$

Solution

$\fbox{B}$

See also

1993 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 1
Followed by
Problem 2
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 26 27 28 29 30
All AHSME Problems and Solutions

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