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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?

Answer :

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 $${cm}^2$$