Question

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

Answer 1

Hint: First, we know from area A of the square as $A=\dfrac{1}{2}\times {{d}^{2}}$. Then, the region I and II are similar from the figure and get the half of total value of square for region I and II. Then, by using the heron’s formula for the region III that is triangle as $A=\sqrt{s\left( s-a \right)\left( s-b \right)\left( s-c \right)}$ to get the region III.

Complete step-by-step answer:
In this question, we are supposed to find how much paper of each shade has been used to make the kite with a shape of square with a diagonal 40 cm and isosceles triangle of base 10 cm and equal sides 13 cm each.

Now, from the given figure we know that the shape of the kite is square with diagonal(d) as 40 cm.
Now, we know from area A of the square that:
$A=\dfrac{1}{2}\times {{d}^{2}}$
So, by substituting the value of diagonal as 40 cm, we get:
$\begin{align}
  & A=\dfrac{1}{2}\times {{\left( 40 \right)}^{2}} \\
 & \Rightarrow A=\dfrac{1}{2}\times 1600 \\
 & \Rightarrow A=800 \\
\end{align}$
So, the area of the kite with region I and II is $800c{{m}^{2}}$.
Now, the region I and II are similar from the figure.
So, the area of region I and II each is:
$\dfrac{800}{2}=400$
So, the area of the region I and II is 400 cm square each.
Now, by using the heron’s formula for the region III that is triangle as:
$\begin{align}
  & s=\dfrac{13+13+10}{2} \\
 & \Rightarrow s=\dfrac{36}{2} \\
 & \Rightarrow s=18 \\
\end{align}$
Now, to get the area of the region III, we get:
$A=\sqrt{s\left( s-a \right)\left( s-b \right)\left( s-c \right)}$
Now, substitute the value of s as calculated above and values of a, b and c from the sides of triangle as:
$\begin{align}
  & A=\sqrt{18\left( 18-13 \right)\left( 18-13 \right)\left( 18-10 \right)} \\
 & \Rightarrow A=\sqrt{18\times 5\times 5\times 8} \\
 & \Rightarrow A=\sqrt{3600} \\
 & \Rightarrow A=60 \\
\end{align}$
So, the area of the region III is 60 cm square.
Hence, the area of the paper required for region I and II is $400c{{m}^{2}}$ each and area of paper required for region III is $60c{{m}^{2}}$.

Note:Now, to solve these types of questions we need to know some of the basic formulas of the figures and the heron’s formula. So, the basic formulas are as:
Area of the square with diagonal d as $A=\dfrac{1}{2}\times {{d}^{2}}$
Area of the triangle by heron’s formulas is as $A=\sqrt{s\left( s-a \right)\left( s-b \right)\left( s-c \right)}$.