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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?
image


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.

image

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 \({cm}^2\)

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