A kite in the shape of a square with a diagonal 32 cm and an isosceles triangle of base 8 cm and sides 6 cm each is to be made of three different shades as shown in the figure. How much paper of each shade has been used in it ?

Explanation

The given figure has been divided into three regions I, II and III consisting of `Delta`ABC, `Delta`ADC and `Delta`DEF respectively.
Join BD, cutting AC at O.
We know that the diagonals of a square are equal and bisect each other at right angles.
`therefore" "AC = BD = 32 cm, OB = OD = 16 cm`,
`angle AOB = angle AOD = 90^(@)`.
Area of Shade I Area `(Delta ABC)=((1)/(2) xx AC xx OB)`
`=((1)/(2) xx 32 xx 16)cm^(2) = 256 cm^(2)`.
Area of Shade II Area `(Delta ACD)=((1)/(2) xx AC xx OD)`
`=((1)/(2)xx 32 xx 16)cm^(2) = 256 cm^(2)`.
Area of Shade III In `Delta`DEF, a = 8 cm, b = 6 cm and c = 6 cm.
`therefore" "s=(1)/(2)(8+6+6)`cm = 10 cm.
`therefore" "`(s - a) = (10 - 8)cm = 2 cm, (s - b) = (10 - 6)cm = 4 cm and (s - c) = (10 - 6)cm = 4 cm.
`therefore" ""area"(DeltaDEF)=sqrt(s(s-a)(s-b)(s-c))`
`=sqrt(10 xx 2 xx 4 xx 4)cm^(2) = 8 sqrt(5)cm^(2)`
`=(8 xx 2.236)cm^(2) ~~ (8 xx 2.24)cm^(2)`
`= 17.92 cm^(2)`.
Hence, the areas of Shade I, Shade II and Shade III are respectively 256 `cm^(2)`, 256 `cm^(2)` and 17.92 `cm^(2)`.