Question

A design is made on a rectangular tile of dimensions 50cm and 70 cm as shown in the figure. The design shows 8 triangles, each of sides 26cm, 17cm and 25cm as cut out. Find the total area of design in $ c{{m}^{2}} $ .

Answer 1

Hint: First, from the figure, we can see that the area A of the rectangular tile is to be calculated first to get the area of the design as $ A=l\times b $ . Then, the other figure in the diagram are 8 identical triangles, so the area of the triangle can be found by using the heron’s formula as $ A=\sqrt{s\left( s-a \right)\left( s-b \right)\left( s-c \right)} $ .Then, the 8 triangles area is cut out from the area of the rectangular tile to get the design area.

Complete step-by-step answer:
In this question, we are supposed to find the total area of the design in the shape of rectangular tile of dimensions 50 cm and 70 cm from which 8 identical triangles are cut with sides 25cm, 17cm and 26cm.

Now, from the figure, we can see that the area A of the rectangular tile is to be calculated first to get the area of the design as:
 $ A=l\times b $
Now, by substituting the value of l as 70 cm and b as 50 cm, we get:
 $ \begin{align}
  & A=70\times 50 \\
 & \Rightarrow A=3500c{{m}^{2}} \\
\end{align} $
So, we get the area of the rectangular tile as $ 3500c{{m}^{2}} $ .
Now, the other figures in the diagram are 8 identical triangles, so the area of the triangle can be found by using the heron’s formula.
Then, we require a term s in the heron’s formula where a, b and c are the sides of the triangle as:
 $ s=\dfrac{a+b+c}{2} $
Then, by substituting the value of a as 25, b as 26 and c as 17, we get:
 $ \begin{align}
  & s=\dfrac{25+26+17}{2} \\
 & \Rightarrow s=\dfrac{68}{2} \\
 & \Rightarrow s=34 \\
\end{align} $
Now, by using the heron’s formula, we get the area A as:
 $ A=\sqrt{s\left( s-a \right)\left( s-b \right)\left( s-c \right)} $
Now, we can substitute all the values to the area of the triangle as:
 $ \begin{align}
  & A=\sqrt{34\left( 34-25 \right)\left( 34-26 \right)\left( 34-17 \right)} \\
 & \Rightarrow A=\sqrt{34\times 9\times 8\times 17} \\
 & \Rightarrow A=\sqrt{41616} \\
 & \Rightarrow A=204 \\
\end{align} $
So, the area of the triangle is $ 204c{{m}^{2}} $ .
Similarly, the area of the 8 such triangles are as:
 $ 204\times 8=1632 $
So, the total area of 8 triangles is $ 1632c{{m}^{2}} $ .
Now, the 8 triangles area is cut out from the area of the rectangular tile to get the design area as:
 $ 3500-1632=1868 $
So, the area of the design cut out is $ 1868c{{m}^{2}} $ .

Note: Now, to solve these type of the questions we need to know some of the basic formula for the area of the triangle as the given triangle is scalene so the most appropriate way we can use is to use the heron’s formula where s is required with formula as $s=\dfrac{a+b+c}{2}$. And then the formula for the area is $A=\sqrt{s\left( s-a \right)\left( s-b \right)\left( s-c \right)}$. Moreover, we can use the other formula for the calculation of the area of the triangle where b is base and height is height as $A=\dfrac{1}{2}\times b\times h$. But, for this we need to make a lot of assumptions that is why heron’s formula is a better option to use.