The area of the white square is 64 square units. The diagonal of the square was extended to point A until the length of segment AB was double the length of the diagonal of the square. What is the area of the gray part?

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

Answer:

64 square units

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