Question

A kite in the shape of a square with a diagonal $32cm$ and an isosceles triangle of base $8cm$ and sides $6cm$ each is to be made of three different shades as shown in Fig. How much paper of each shade has been used in it?

Answer 1

According to the question,

Let the kite is made with square$ABCD$ & an isosceles triangle$DEF$.

Given, sides of a triangle $DEF$ are $DE$ = $DF$ = $6cm$ & $EF$ = $8cm$

& diagonals of a square $ABCD=32cm$

We know that,

As the diagonal of a square bisect each other at the right angle.

$OA=OB=OC=OD=32/12=16$ cm

$AO$ perpendicular $BC$ & $DO$ perpendicular $BC$

Area of region $I$ = Area of $\Delta ABC=\frac{1}{2}\times BC\times OA$

Area of the region $I$ $=\frac{1}{2}\times 32\times 16=256c{m}^2$

Similarly the area of region $II$ $=256c{m}^2$

For the section $III$

Now, in $\Delta DEF$

Let the side a = 6 cm, b = 6 cm, c = 8 cm

Semi perimeter of a triangles $=(6+6+8)/2$ cm $=10$ cm

Are of the III triangle piece

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

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

$=\sqrt{2\times 5\times 4\times 4\times 2}$

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

$=2\times 4\sqrt{5}=8\sqrt{5}$

$8\times 2.24=17.92c{m}^2.....$ ($\sqrt{5}=2.24$)

Hence area of paper of $I$ color used in making kite $=256c{m}^2$

Area of paper of $II$ color used in making kite $=256c{m}^2$

And area of paper of $III$ color in making kite $=17.92c{m}^2$