Difference between revisions of "1979 AHSME Problems/Problem 12"

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Solution by e_power_pi_times_i
 
Solution by e_power_pi_times_i
  
Because <math>AB = OD</math>, triangles <math>ABO</math> and <math>BOE</math> are isosceles. Denote <math>\measuredangleBAO = \measuredangleAOB = \theta</math>. Then <math>\measuredangleABO = 180^\circ-2\theta</math>, and <math>\measuredangleEBO = \measuredangleOEB = 2\theta</math>, so <math>\measuredangleBOE = 180^\circ-4\theta</math>. Notice that <math>\measuredangleAOB + \measuredangleBOE + 45^\circ = 180^\circ</math>. Therefore <math>\theta+180-4\theta = 135^\circ</math>, and <math>\theta = \boxed{\textbf{(B) } 15^\circ}</math>.
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Because <math>AB = OD</math>, triangles <math>ABO</math> and <math>BOE</math> are isosceles. Denote <math>\measuredangle BAO = \measuredangle AOB = \theta</math>. Then <math>\measuredangle ABO = 180^\circ-2\theta</math>, and <math>\measuredangle EBO = \measuredangle OEB = 2\theta</math>, so <math>\measuredangle BOE = 180^\circ-4\theta</math>. Notice that <math>\measuredangle AOB + \measuredangle BOE + 45^\circ = 180^\circ</math>. Therefore <math>\theta+180-4\theta = 135^\circ</math>, and <math>\theta = \boxed{\textbf{(B) } 15^\circ}</math>.
  
 
== See also ==
 
== See also ==

Revision as of 12:58, 6 January 2017

Problem 12

[asy] size(200); pair A=(-2,0),B,C=(-1,0),D=(1,0),EE,O=(0,0); draw(arc(O,1, 0, 180)); EE=midpoint(arc(O,1, 0, 90)); draw(A--EE); draw(A--D); B=intersectionpoint(arc(O,1, 180, 0),EE--A); draw(O--EE); label("$A$",A,W); label("$B$",B,NW); label("$C$",C,S);label("$D$",D,E);label("$E$",EE,NE);label("$O$",O,S);label("$45^\circ$",(0.25,0.1),fontsize(10pt)); //Credit to TheMaskedMagician for the diagram[/asy]

In the adjoining figure, $CD$ is the diameter of a semi-circle with center $O$. Point $A$ lies on the extension of $DC$ past $C$; point $E$ lies on the semi-circle, and $B$ is the point of intersection (distinct from $E$ ) of line segment $AE$ with the semi-circle. If length $AB$ equals length $OD$, and the measure of $\measuredangle EOD$ is $45^\circ$, then the measure of $\measuredangle BAO$ is

$\textbf{(A) }10^\circ\qquad \textbf{(B) }15^\circ\qquad \textbf{(C) }20^\circ\qquad \textbf{(D) }25^\circ\qquad \textbf{(E) }30^\circ$

Solution

Solution by e_power_pi_times_i

Because $AB = OD$, triangles $ABO$ and $BOE$ are isosceles. Denote $\measuredangle BAO = \measuredangle AOB = \theta$. Then $\measuredangle ABO = 180^\circ-2\theta$, and $\measuredangle EBO = \measuredangle OEB = 2\theta$, so $\measuredangle BOE = 180^\circ-4\theta$. Notice that $\measuredangle AOB + \measuredangle BOE + 45^\circ = 180^\circ$. Therefore $\theta+180-4\theta = 135^\circ$, and $\theta = \boxed{\textbf{(B) } 15^\circ}$.

See also

1979 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 11
Followed by
Problem 13
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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