Q27 of 37 Page 13

In a circular table cover of radius 32cm, a design is formed leaving an equilateral triangle ABC in the middle as shown in the given figure. Find the area of the design (shaded region)


Given: Radius of circle = 32cm


Area of design = Area of circle – Area of ΔABC


Firstly, we find the area of a circle


Area of circle = πr2




…(a)


Now, we will find the area of equilateral ΔABC



Construction:


Draw OD ⊥ BC


In ΔBOD and ΔCOD


OB = OC (radii)


OD = OD (common)


∠ODB = ∠ODC (90°)


∴ ΔBOD ≅ ΔCOD [by RHS congruency]


⇒ BD = DC [by CPCT]


or BC = 2BD …(i)


and,


Now, In ΔBOD, we have




⇒ BD = 16√3 cm


From (i), BC = 2BD ⇒ BC = 32√3 cm


Now, Area of equilateral ΔABC




= 768√3 cm2 …(b)


Therefore, Area of design = Area of circle – Area of ΔABC


[from (a) and (b)]


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