MathematicsIf an isosceles triangle ABC, in which AB=AC=6 cm,   is inscribed in a circle of radius 9 cm.   find the area of the triangle.

If an isosceles triangle ABC, in which AB=AC=6 cm,   is inscribed in a circle of radius 9 cm.   find the area of the triangle.


  1. A
    4 2   cm 2  
  2. B
    4 3   cm 2  
  3. C
    2 2   cm 2  
  4. D
    2 3   cm 2   

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    Solution:

    Given that, ABC is isosceles triangle, AB=AC=6 cm,   inscribed in circle of radius 9cm.
    Drawing the triangle inscribed in a circle,
    P is the mid-point of BC, because ΔABC   is an isosceles triangle.
    Let AP=x,BP=PB=y  .
    Applying Pythagoras theorem in ΔABP  ,
    6 2 = x 2 + y 2  
    36 x 2 = y 2  ……….(1)
    Applying Pythagoras theorem in ΔBPO  ,
    O B 2 =O P 2 +B P 2  
    9 2 = (9x) 2 + y 2  .(2)
    Substitute 36 x 2 = y 2   in (2),
    81= (9x) 2 +36 x 2  
    81=36+81+ x 2 18x+ x 2 0=3618x 18x=36  
    x=2  
    Substituting the value of x in equation (1),
    36 2 2 = y 2 y 2 =364 y= 32 y=4 2  
    Now,
    Area of triangle = 1 2 ×Base×Height   Area= 1 2 (2)(4 2 )   Area=4 2   cm 2  
    Therefore, option (1) is correct.
     
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