Answer :
Ratio of the area of a square inscribed in a semicircle with radius 'r' to the area of a square inscribed in a circle with radius 'r' is equal to 2: 5.
As given in the question,
Let ABCD be the square with side 'a' inscribed in a semicircle with radius 'r'.
'O' be the center of the semicircle.
'M' be the midpoint of AB such that MB = a/2
Join OM, OB and OM ⊥ AB
OB = r
OM = BC = a
In ΔOMB.
OB² = OM² + MB²
⇒ r² = a² + (a/2)²
⇒r² = 5a²/4
⇒a² = 4r²/5
Area of the square inscribed in a semicircle = a²
= 4r²/5
Let PQRS be the inscribed in a circle with side 'x'
Diagonal of the square inscribed in a circle = 2r
In ΔPQR,
PR = diagonal
= 2r
PQ = QR = x
PR² = PQ² + QR²
⇒(2r)² = x² + x²
⇒x² = 4r² /2
⇒x² = 2r²
Area of the square inscribed in a circle = x²
= 2r²
Ratio of the area of the square inscribed in ( semicircle to the : circle )
= 4r²/5 / 2r²
= 2 / 5
= 2 : 5
Therefore, ratio of the area of the squares inscribed in a semicircle to the circle is 2 : 5.
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