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 \(A\)?

Logic Jun 9, 2019

#1**0 **

*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 75^{o} 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).

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Guest Jun 10, 2019

#2**+2 **

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"

CPhill Jun 11, 2019