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 Fig. 12.17. How much paper of each shade has been used in it?

We know that, Heron's formula

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

where, s is the semi-perimeter = half of the perimeter

and  a, b and c are the sides of the triangle 

Given that, A kite in the shape of a square with a diagonal 32 cm

Given diagonal BD = AC = 32 cm, then OA = 1/2 AC = 16 cm.

So square ABCD is divided into two isoscales triangles ABD and CBD of base 32 cm and height 16 cm.

Area of ∆ABD = 1/2  $\times$base $\times$ height

= (32 $\times$ 16)/2

= 256 cm²

 Therefore, Area of ∆ABD = Area of ∆CBD = 256 cm²

Now, for ∆CEF

Semi Perimeter = s = (a $+$ b $+$c)/2

s = (6 $+$ 6 $+$ 8)/2

s = 20/2

s = 10 cm

By using above formula

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

= $\sqrt{10(10 - 6)(10 - 6)(10 - 8)}$

= $\sqrt{10 \times 4 \times 4 \times 2}$

= 8 $\sqrt{5}$

= 8 $\times$ 2.24

= 17.92 cm²

Hence the area of the paper used to make region I = 256 cm²,

 region II = 256 cm²,

region III = 17.92 cm².