Question

A kite in the shape of a square with a diagonal $40\text{cm}$ and an isosceles triangle of base $10\text{cm}$ and equal sides $13\text{cm}$ each, is to be made up of three different shades as shown in the figure. How much paper of each shade has been used to make the kite?

Answer 1

Given, diagonal of the square$=40\text{cm}$

If $a$ is the side of the square, then:

$\sqrt{a^2+a^2}=40$

$a=20\sqrt{2}\text{cm}$

The square is divided into $4$ parts, $2$ of which are white and $2$ are dark.

Area of each quadrant of square $=\frac{1}{4}\times \text{Area of square}$

$=\frac{1}{4}\times 20\sqrt{2}\times 20\sqrt{2}$

$=200{\text{cm}}^2$

Area of white paper used in the kite$=2\times 200{\text{cm}}^2=400{\text{cm}}^2$

Calculate the area of the isosceles triangle at the bottom part using Heron's formula:

$a=13\text{cm},b=13\text{cm},c=10\text{cm}$

$s=\frac{a+b+c}{2}=18\text{cm}$

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

$=\sqrt{18(18-13)(18-13)(18-10)}$

$=60{\text{cm}}^2$

Area of darker paper used$=$ Area of 2 quadrants of the square $+$ Area of isosceles triangle

$=(400+60){\text{cm}}^2$

$=460{\text{cm}}^2$