Question

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 1

64 square units

Answer 2

Area of the gray region = 64 square units

Explanation:

Area of the white square = 64 square units

Length of a side of the given square =
`\sqrt{\text{Area}}`

=
`√(64)`

= 8 units

By Pythagoras theorem,

Length of diagonal DB =
`√(BC^2+CD^2)`

DB =
`√(8^2+8^2)`

=
`8√(2)`

AD = DB =
`8√(2)` [Given]

OD = OC =
`(1)/(2)(DB)`

= 4√2

Therefore, AO = AD + OD = 8√2 + 4√2

= 12√2

Area of ΔACD = Area of ΔAOC - Area of ΔCOD

=
`(1)/(2)(AO)(OC)-(1)/(2)(CO)(OD)`

=
`(1)/(2)[(12√(2)* 4√(2))-(4√(2))^2]`

=
`(1)/(2)[96-32]`

= 32

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

= 32 + 32

= 64 square units