A very beautiful question. you would love solving it.
What part of larger square is shaded?
Answers
From image we have :-
→ ABCD is a Square = with Side Let 2a cm.
→ EFGH are the Mid - Points of Sides of Square.
→ 4 Quadrant are Drawn from Each vertex of Square .
→ In Middle A Square IJKL is Drawn with shaded Portion .
Solution :-
→ sides of Square = 2a cm.
→ Diagonal of Square = √2 * side = 2√2a cm = AC .
Now,
→ AS = AH = AL = Radius of Quadrant = a cm.
→ CG = CF = CJ = Radius of Quadrant = a cm.
And,
→ Diagonal AC = AL + LJ + JC
Putting value we get :-
→ 2√2a = a + LJ + a
→ 2√2a = 2a + LJ
→ LJ = 2√2a - 2a
→ LJ = 2a[√2 - 1] = Diagonal of Square IJKL.
_________________
Hence,
☛ Area of Square IJKL = (1/2) * (Diagonal)²
☛ Area[IJKL] = (1/2) * [2a(√2-1)]²
☛ Area[IJKL] = (1/2) * [ 4a²(√2 - 1)² ]
☛ Area[IJKL] = 2a² (2 + 1 - 2√2)
☛ Area[IJKL] = 2a²(3 - 2√2) .
☛ Area[IJKL] = 2a²( 3 - 2*1.414)
☛ Area[IJKL] = 2a² * (0.171)
☛ Area[IJKL] = (0.343)a² = Shaded Area.
_________________
Now,
→ Area of Square ABCD = (side)² = (2a)² = 4a².
Hence,
☞ Part of larger square is shaded = (Shaded Area) / (Area of Square ABCD).
☞ Part of larger square is shaded = (0.343)a²/(4a²)
☞ Part of larger square is shaded = (1/12) [ Approx] . (Ans).
Hence, (1/12) Part of larger square is shaded.
The question is indeed beautiful... and interesting...
Solution in the attachment....
