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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
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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"

Jun 11, 2019
edited by CPhill  Jun 11, 2019
edited by CPhill  Jun 11, 2019
edited by CPhill  Jun 11, 2019
#3
+105683
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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
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THX, Melody  !!!!

CPhill  Jun 20, 2019