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# 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 figure. How much paper of each shade has been used in it?

Since, the kite is in the shape of a square.

Each diagonal of square = 32 cm ...(Given)

We know that, the diagonals of a square bisect each other at right angle.

For Part I :
Area of Part I
= $$\frac{1}{2}$$ × Base × Height
= $$\frac{1}{2}$$ × 32 × 16
= 16 × 16 = 256$${cm}^2$$

For Part II :
Area of Part II
= $$\frac{1}{2}$$ × Base × Height
= $$\frac{1}{2}$$ × 32 × 16
= 16 × 16 = 256$${cm}^2$$

For Part III :
It is a triangle with sides 6 cm, 6 cm and 8 cm.
Now, we know that,

$$s = \frac{a + b + c}{2}$$
$$\therefore$$ $$s = \frac{6 + 6 + 8}{2} = \frac{20}{2} = 10cm$$

Now, Area of triangle
= $$\sqrt{10(10 - 6)(10 - 6)(10 - 8)}$$
(Since, Heron's formula [area = $$\sqrt{s(s - a)(s - b)(s - c)}$$])
= $$\sqrt{10 × 4 × 4 × 2}$$

= $$\sqrt{5 × 2 × 4 × 4 × 2}$$
= $$2 × 4\sqrt{5} {cm}^2$$
= $$8\sqrt{5} {cm}^2$$
= 8 × 2.24 cm = 17.92 $${cm}^2$$

Hence, the area of colour used for Paper I = 256 $${cm}^2$$

The area of colour used for Paper II = 256 $${cm}^2$$

The area of colour used for Paper III = 17.92 $${cm}^2$$