# Find the shortest distance from ( 0,0 ) to ( 12,12) without going inside the circle with centre ( 6,6 ) radius 5.

2
by 99559955

2014-12-08T16:31:22+05:30

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See diagram.

Let D be (12, 12)

Draw y = x line. Draw the circle  (x-6)^2 + (y-6)^2 = 5^2

The intersection points of these two are  A and B.

Hence,     2 (x-6)^2 = 25
x = 6 +  5 / √2  (point B)   or   6 - 5 / √2 (point A)
y = x

We want the total of distance OA straight line distance, AEFB (semicircle perimeter, and straight line distance BD.

OA = √2 * (6-5/√2)
AEFB = π * 5
BD = √2 * (6-5/√2),     (OA and BD are equal due to symmetry.

Hence the Shortest distance = 12 √2 - 10 + 5 π

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2014-12-09T16:09:52+05:30
Hence the shortest distance = 12√2-10+5π