Mathematics: 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?

The area of the white square is 64 square units. The

rodfam13716

07.04.2022

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 = $\sqrt{\text{Area}}$

= $\sqrt{64}$

= 8 units

By Pythagoras theorem,

Length of diagonal DB = $\sqrt{BC^2+CD^2}$

DB = $\sqrt{8^2+8^2}$

= $8\sqrt{2}$

AD = DB = $8\sqrt{2}$ [Given]

OD = OC = $\frac{1}{2}(DB)$

= 4√2

Therefore, AO = AD + OD = 8√2 + 4√2

= 12√2

Area of ΔACD = Area of ΔAOC - Area of ΔCOD

= $\frac{1}{2}(AO)(OC)-\frac{1}{2}(CO)(OD)$

= $\frac{1}{2}[(12\sqrt{2}\times 4\sqrt{2})-(4\sqrt{2})^2]$

= $\frac{1}{2}[96-32]$

= 32

Therefore, area of gray part = Area of ΔACD + Area of ΔAED

= 32 + 32

= 64 square units

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