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The figure below shows a triangle ABC inscribed in a circle. AC = 10cm, BC = 7cm and AB = 10cm.
MathF4OPP1Q20

  1. Find the size of angle BAC. 
  2. Find the radius of the circle. 
  3. Hence calculate the area of the shaded region.

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  1. a2 =b2+ c2 – 2bc cos A
    72 = 10+ 82 - (2 x 10 x 8) cos A
    49=164- 60 cos A
    -116 =160 cos A
    Cos A =116/160
    Cos A = 0.725
    Cos1 0.725 = 43.53115
    <BAC = 43.53
     
  2. a/(sin sin A ) = 2R
    7/(sin 43.53) = 2R
    R =5.082cm
     
  3. Sin 43.53 = 3.5/r
    r = 3.5/(sin sin 43.53 )= 5.082 cm
    Area of △OCB =1/2 ab sin θ
    = 1/2 x 5.082 sin 87.06
    = 12.896cm2
    Area of sector ACB
    =θ/360 πr2
    =87.06/360 22/7 x 5.082 = 19.630
    Shaded region
    (19.630 — 12.896) = 6.734 cm2
     
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