Difference between revisions of "1993 AHSME Problems/Problem 23"

(Created page with "== Problem == <asy> draw(circle((0,0),10),black+linewidth(.75)); draw((-10,0)--(10,0),black+linewidth(.75)); draw((-10,0)--(9,sqrt(19)),black+linewidth(.75)); draw((-10,0)--(9,-s...")
 
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== Solution ==
 
== Solution ==
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We have all the angles we need, but most obviously, we see that right angle in triangle <math>ABD</math>.
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Note also that angle <math>BAD</math> is 6 degrees, so length <math>AB = cos(6)</math> because the diameter, <math>AD</math>, is 1.
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Now, we can concentrate on triangle <math>ABX</math> (after all, now we can decipher all angles easily and use Law of Sines).
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We get:
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<math>AB  /\angle{AXB} = AX / \angle{ABX}</math>
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That's equal to
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<math>cos(6)/ sin(180-18) = AX / sin 12</math>
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Therefore, our answer is equal to:
 
<math>\fbox{B}</math>
 
<math>\fbox{B}</math>
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Note that <math>sin(162) = sin(18)</math>, and don't accidentally put <math>\fbox{B}</math> because you thought 1/sin was sec!
  
 
== See also ==
 
== See also ==

Revision as of 20:40, 26 June 2017

Problem

[asy] draw(circle((0,0),10),black+linewidth(.75)); draw((-10,0)--(10,0),black+linewidth(.75)); draw((-10,0)--(9,sqrt(19)),black+linewidth(.75)); draw((-10,0)--(9,-sqrt(19)),black+linewidth(.75)); draw((2,0)--(9,sqrt(19)),black+linewidth(.75)); draw((2,0)--(9,-sqrt(19)),black+linewidth(.75)); MP("X",(2,0),N);MP("A",(-10,0),W);MP("D",(10,0),E);MP("B",(9,sqrt(19)),E);MP("C",(9,-sqrt(19)),E); [/asy]

Points $A,B,C$ and $D$ are on a circle of diameter $1$, and $X$ is on diameter $\overline{AD}.$

If $BX=CX$ and $3\angle{BAC}=\angle{BXC}=36^\circ$, then $AX=$


$\text{(A) } cos(6^\circ)cos(12^\circ)sec(18^\circ)\quad\\ \text{(B) } cos(6^\circ)sin(12^\circ)csc(18^\circ)\quad\\ \text{(C) } cos(6^\circ)sin(12^\circ)sec(18^\circ)\quad\\ \text{(D) } sin(6^\circ)sin(12^\circ)csc(18^\circ)\quad\\ \text{(E) } sin(6^\circ)sin(12^\circ)sec(18^\circ)$

Solution

We have all the angles we need, but most obviously, we see that right angle in triangle $ABD$.

Note also that angle $BAD$ is 6 degrees, so length $AB = cos(6)$ because the diameter, $AD$, is 1.

Now, we can concentrate on triangle $ABX$ (after all, now we can decipher all angles easily and use Law of Sines).

We get:

$AB  /\angle{AXB} = AX / \angle{ABX}$

That's equal to

$cos(6)/ sin(180-18) = AX / sin 12$

Therefore, our answer is equal to: $\fbox{B}$

Note that $sin(162) = sin(18)$, and don't accidentally put $\fbox{B}$ because you thought 1/sin was sec!

See also

1993 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 22
Followed by
Problem 24
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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