Solution
Given:• ABCD is a square with side = 1
• AFC is an arc, centered at B
• AEC is an arc, centered at D
To find:• Area of the rhombus AECF
Approach and Working: • To know the area of a rhombus, we need to find the length of its diagonals, as area of a rhombus = \(\frac{1}{2}\) * product of the diagonals
• For square ABCD with side 1-unit, diagonal AC = \(\sqrt{2}\)
o AC is also the longer diagonal of AECF
• Another diagonal EF = BD – BE – DF
o Now, DE = 1, as the radius of the semicircle is equal to the side of the square
o Therefore, BE = BD – DE = \(\sqrt{2} – 1\)
o Similarly, DF = \(\sqrt{2} – 1\)
o Hence, EF = \(\sqrt{2} – 2 * (\sqrt{2} – 1) = 2 – \sqrt{2}\)
o Area of the rhombus AECF = \(\frac{1}{2} * AC * EF = \frac{1}{2} * \sqrt{2} * 2 – \sqrt{2} = \sqrt{2} – 1\)
Hence, the correct answer is option B.
Answer: B _________________