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Stuart has drawn a pair of concentric circles, as shown. He draws chords \(\overline{AB}\),\(\overline{BC}, \ldots\)  of the large circle, each tangent to the small one. If \(m\angle ABC=75^\circ\), then how many segments will he draw before returning to his starting point at \(A\)?

 Jun 9, 2019
 #1
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how many segments will he draw before returning to his starting point at A 

 

Nobody has answered this yet, so I'm going to take a run at it. 

 

Never mind the inside circle; we don't need it.  Just draw each successive chord at a 75o angle.  I went ahead and starting drawing them, and the end of the tenth chord landed at A again.  Stuart will draw a total of 10 lines (i.e., an additional 7 to the 3 that were already there).

 

.

 Jun 10, 2019
 #2
avatar+102943 
+3

Look at the following image:

 

I agree with the guest....the inner circle is not needed

In the circle,  the angles E and D are bisected by OE and OD....so the measure of angles OED and ODE = 37.5°....so angle DOE = 105°

So....each chord will span an arc of 105°

 

Note that.... 

 

3 chords span 3(105) = 315°  = - 45°    and we are at point "D"

6 chords span 6(105)  = 630° = - 90°   and we are at point "G"

 

So....every 3 chords will end up shifting us  a multiple of -45° from "A" 

 

So...we need      360 / 45 * (3)  =  8 * 3  = 24  chords to bring us back to "A"

 

 

cool cool cool

 Jun 11, 2019
edited by CPhill  Jun 11, 2019
edited by CPhill  Jun 11, 2019
edited by CPhill  Jun 11, 2019
 #3
avatar+103674 
+1

That is really cool Chris,

I had a quick look at it when it posted but I didn't make any headway.

 

I admit I have not studied your answer yet but I am going to put it into the feature questions post because I want to look at it properly when I get a chance and I think some other people would like to too.

Melody  Jun 19, 2019
 #4
avatar+102943 
+1

THX, Melody  !!!!

 

 

cool cool cool

CPhill  Jun 20, 2019

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