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In the figure above, the circles are concentric with centre O. B is a point on the larger circle. D is a point on the smaller circle. BD is joined, provided that BD touches the circle at only one point. Now OB is joined and extended to meet the larger circle at A. Find the distance between points A and D if the radii of the circles are 4*sqrt(13)​ and 8 units respectively.

 

 Jul 29, 2020
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This is probably not the most efficient way, but it works.

 

Let O = (0, 0), D = (0, -8). Then B = (-12, -8) because OD = 8 and OB = 4sqrt(13), so by the Pythagorean Theorem, BD = 12

 

Then, since AB is a line segment passing through O, we have that A = (12, 8). Use the distance formula from (0, -8) to (12, 8) and get sqrt((12-0)^2+(8-(-8))^2) = sqrt(12^2+16^2) = 20, so AD = 20.

 

You are very welcome!

:P

 Jul 29, 2020

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