A circle has diameter AD of length 400.B and C are points on the same arc of AD such that |AB|=|BC|=60.What is the length |CD|?

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Jaida Sanders · Accepted Answer

We describe a trigonometric approach. We will use a couple of times the following fact. Let   △  P  Q  R be isosceles, with   P  Q  =  P  R  =  a and   Q  R  =  b. Let   y  =  ∠  Q  P  R. Then             b              /            2        a    =  sin  ⁡  (  y      /    2  ).This is easily proved by joining   P to the midpoint   M of   Q  R.Now let O be the centre of the circle, and let   ∠  A  O  B  =  x. Then   ∠  B  O  C  =  x. and   ∠  C  O  D  =      180    ∘    −  2  x. It follows that half of   ∠  C  O  D is       90    ∘    −  x.By basic trigonometry, the length of   C  D is   200  sin  ⁡  (      90    ∘    −  x  ), that is,   200  cos  ⁡  x.It rmains to find   cos  ⁡  x. From   △  A  O  B, we see that   sin  ⁡  (  x      /    2  )  =      30    200    =      3    20  .But by a double angle identity, we have  cos  ⁡  x  =  1  −  2      sin    2    ⁡  (  x      /    2  )  ,and now we know   cos  ⁡  x.

A circle has diameter AD of length 400. B and C

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