Answer:
Area of the gray region = 64 square units
Step-by-step explanation:
Area of the white square = 64 square units
Length of a side of the given square = [tex]\sqrt{\text{Area}}[/tex]
= [tex]\sqrt{64}[/tex]
= 8 units
By Pythagoras theorem,
Length of diagonal DB = [tex]\sqrt{BC^2+CD^2}[/tex]
DB = [tex]\sqrt{8^2+8^2}[/tex]
= [tex]8\sqrt{2}[/tex]
AD = DB = [tex]8\sqrt{2}[/tex] [Given]
OD = OC = [tex]\frac{1}{2}(DB)[/tex]
= 4√2
Therefore, AO = AD + OD = 8√2 + 4√2
= 12√2
Area of ΔACD = Area of ΔAOC - Area of ΔCOD
= [tex]\frac{1}{2}(AO)(OC)-\frac{1}{2}(CO)(OD)[/tex]
= [tex]\frac{1}{2}[(12\sqrt{2}\times 4\sqrt{2})-(4\sqrt{2})^2][/tex]
= [tex]\frac{1}{2}[96-32][/tex]
= 32
Therefore, area of gray part = Area of ΔACD + Area of ΔAED
= 32 + 32
= 64 square units
