Q.149. Elementary Trigonometry.

Question 149.

Find the length AD in the given figure.

Answer 149.
(Refer to the figure below.)

Drop a line from O' to C.

Draw a perpendicular ON from O on O'C.
O'C = O'N + NC
=> R = (R+r)sint + r
=> sint = (R-r) / (R+r).
(r + R) sint + r = R
=> sint = (R - r) / (R + r)
=> cost
= √[1 - (R - r)^2 / (R + r)^2]
= 2√(rR)/(R+r)

=> AD
(r  + R) (1 + cost)
= (r + R) * [1 + 2√(rR)/(R+r)]
= r + R + 2√(rR).

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