# Q7.    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?

From the figure:

Calculation of the area for each shade:

Here, base $BD = 32\ cm$ and the height $AO = 16\ cm.$

Therefore, the area of triangle ABD will be:

$= \frac{1}{2} \times base\times height = \frac{1}{2}\times 32\times 16$

$= 256\ cm^2$

Hence, the area of paper used in shade I is $256\ cm^2.$

Here, base $BD = 32\ cm$  and height $CO = 16\ cm.$

Therefore, the area of triangle CBD will be:

$= \frac{1}{2} \times base\times height = \frac{1}{2}\times 32\times 16$

$= 256\ cm^2$

Hence, the area of paper used in shade II is $256\ cm^2.$

Here, the sides are of lengths, $a = 6\ cm,\ b = 6\ cm\ and\ c = 8\ cm.$

So, the semi-perimeter of the triangle:

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

Therefore, the area of the triangle can be found by using Heron's formula:

$Area = \sqrt{s(s-a)(s-b)(s-c)}$

$= \sqrt{10(10-6)(10-6)(10-8)}$

$= \sqrt{10(4)(6)(2)}$

$= 8\sqrt{5}\ cm^2$

Hence, the area of the paper used in shade III is $8\sqrt{5}\ cm^2$.

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