Question

An umbrella is made by stitching 12 triangular pieces of cloth. Each piece measures 50cm, 20cm and 50cm, find the area of the cloth used in it.

A. $2400\sqrt 6 c{m^2}$
B. $2200\sqrt 3 c{m^2}$
C. $1200\sqrt 6 c{m^2}$
D. $1000\sqrt 6 c{m^2}$

Answer 1

Hint: Given that umbrella is made of 12 triangular pieces of cloth. The area of the triangle when three sides are given is $\sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} $, where $s = \dfrac{{a + b + c}}{2}$ ; a, b, c are the three sides of the triangle. Find the area of one triangular piece and multiply it with 12 to get the total area of the cloth used.

Complete step-by-step answer:
We are given that an umbrella is made by stitching 12 triangular pieces of cloth and each piece measures 50cm, 20cm and 50cm.
We have to calculate the total area of cloth used to make the umbrella.

Area of one triangular piece is $\sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} $, where $s = \dfrac{{a + b + c}}{2}$ a=50cm, b=20cm and c=50cm
$
  s = \dfrac{{50 + 20 + 50}}{2} \\
  \to s = \dfrac{{120}}{2} \\
  \therefore s = 60cm \\
 $
Area of triangular piece is
$
  Area = \sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \\
  s = 60,a = 50,b = 20,c = 50 \\
  \to Area = \sqrt {60\left( {60 - 50} \right)\left( {60 - 20} \right)\left( {60 - 50} \right)} \\
  \to Area = \sqrt {60 \times 10 \times 40 \times 10} \\
  \to Area = \sqrt {24 \times {{10}^4}} = \sqrt {4 \times 6 \times {{10}^4}} \\
  \to Area = 2 \times {10^2}\sqrt 6 = 200\sqrt 6 c{m^2} \\
 $
Total area of the cloth used is $12 \times 200\sqrt 6 = 2400\sqrt 6 c{m^2}$
 Therefore, the correct option is Option A, $2400\sqrt 6 c{m^2}$
So, the correct answer is “Option A”.

Note: Another approach
Area of the triangle = $\dfrac{1}{2}(b \times h)$ , where ‘b’ is the base and ‘h’ is the height of the triangle
So, height is not given in the question which means we have to find the height of the triangle using the given three sides’ measurements.
The given triangle is an isosceles triangle so the height will divide it into two smaller congruent triangles and the base will also be bisected.
   

As we can see the diagram, the height divided the triangle into two right triangles.
Using Pythagoras theorem, we can find the height.
According to Pythagoras theorem, the hypotenuse square is equal to the sum of squares of the other two adjacent sides.
$
  {c^2} = {\left( {\dfrac{b}{2}} \right)^2} + {h^2} \\
  c = 50,b = 20,\dfrac{b}{2} = 10 \\
  {50^2} = {10^2} + {h^2} \\
  {h^2} = 2500 - 100 = 2400 \\
  h = \sqrt {2400} = 20\sqrt 6 cm \\
 $
Area is
 $
  \dfrac{1}{2} \times b \times h \\
  b = 20cm,h = 20\sqrt 6 cm \\
  Area = \dfrac{1}{2} \times 20 \times 20\sqrt 6 = 200\sqrt 6 c{m^2} \\
 $
Total area of the cloth used is $12 \times 200\sqrt 6 = 2400\sqrt 6 c{m^2}$