if ABCD is a square and the area covered by the blue equilateral triangle is 49 under root 3 cm square what will be the area covered by the yellow region in cm is square.
a 139
b 98
c 42
d 154
Answers
Solution :-
Given that
ABCD is a square.
The area of the blue equilateral triangle
= 49√3 cm²
We know that
Area of an Equilateral triangle whose side is ' a ' units is (√3/4)a² sq.units.
=> (√3/4)a² = 49√3
=> a² = 49√3×(4/√3)
=> a² = 49/4
=> a = ±√(49/4)
=> a = ±7/2
Therefore, a = 7/2 cm
Since, The length of the side can't be negative.
The side of the equilateral triangle = 7/2 cm
From the figure
The side of the equilateral triangle
= The side of the square
= 7/2 cm
We know that
Area of a square = side × side sq.units
Area of the square = (7/2)×(7/2) cm²
=> Area = (7×7)/(2×2)
=> Area of the square = 49/4 cm²
From the figure,
The side of the square = The diameter of the semi-circle = 7/2 cm
Diameter = 7/2 cm
Radius = Diameter /2
=> Radius = (7/2)/2
=> Radius 7/(2×2)
The Radius of the semi circles (r) = 7/4 cm
We know that
Area of a semi circle is πr²/2 sq.units
Area of the semi circle = (22/7)×(7/4)² cm²
=> Area = [(22/7)×(7/4)×(7/4)]/2
=> Area = (22×7×7)/(7×4×4×2)
=> Area = (11×7)/(4×4)
=> Area = 77/16 cm²
Area of one semi circle = 77/16 cm²
Area of the two semi-circles = 2(77/16) cm²
=> Area of the two semi-circles = 77/8 cm²
From the figure,
Area of the yellow region = Area of the square - Area of two semi circles.
=> (49/4)-(77/8)
LCM of 4 and 8 = 8
=> (98-77)/8
=> 21/8 cm²
=> 2.625 cm²
Answer :-
The area is covered by the yellow region is
21/8 cm² or 2.625 cm²
Used formulae:-
♦ Area of an Equilateral triangle whose side is 'a' units is (√3/4)a² sq.units.
♦ Area of a square = side × side sq.units
♦ Area of a semi circle is πr²/2 sq.units
♦ Radius = Diameter / 2
- a = side
- r = radius
- d = diameter
- π = 22/7