Question

Find the area of a triangle two sides of which are 18 cm and 10 cm and the perimeter is 42 cm.

Answer 1

Hint: Use the Heron’s formula to find the area of the triangle which is given by the formula \[A = \sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \] , where \[s = \dfrac{{a + b + c}}{2}\]
Perimeter of a triangle is the sum of the length of the sides of a triangle, in this question the perimeter and the length of the two sides of the triangle is given so by using the perimeter of the triangle we will find the third side of the triangle and by using these sides in Heron’s formula we will find the area of the triangle.

Complete step-by-step answer:
Let the sides of the triangle whose area is to be calculated be \[a,b,c\]
So we can write
The length of the first side of triangle \[a = 18\;cm\]
The length of the second side of triangle \[b = 10\;cm\]
The length of the third side of triangle \[c = x\]
Now as we know the perimeter of the triangle is the sum of the length of the side of the triangle, hence we can write
 \[P = a + b + c\]
By substituting the values we can write
 \[
  42 = 18 + 10 + x \\
   \Rightarrow x = 42 - 28 \\
   \Rightarrow x = 14\;cm \;
 \]
Hence the length of the third side of triangle \[c = 14cm\]
Now to find the area of the triangle we will use the Heron’s formula which is given as \[A = \sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \]
Here semi-perimeter \[s = \dfrac{{a + b + c}}{2}\] , so by substituting the value of the length of the side of triangle we get
 \[
  s = \dfrac{{18 + 10 + 14}}{2} \\
   = \dfrac{{42}}{2} \\
   = 21\;cm \;
 \]
Hence we get semi perimeter \[s = 21cm\]
Now we substitute the value of \[s\] in Heron’s formula to find the area of the triangle, hence we can write
 \[
  A = \sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \\
   = \sqrt {21\left( {21 - 18} \right)\left( {21 - 10} \right)\left( {21 - 14} \right)} \\
   = \sqrt {21 \times 3 \times 11 \times 7} \\
   = \sqrt {4851} \\
   = 21\sqrt {11} c{m^2} \;
 \]
Hence the area of the triangle is \[ = 21\sqrt {11} c{m^2}\]
So, the correct answer is “ \[ 21\sqrt {11} c{m^2}\] ”.

Note: Heron’s formula to find the area of the triangle requires the length of all sides of the triangle which is one of the demerits of this formula so to us this formula we need to find all the sides of the triangle but this formula can give the area of the triangle without the measurements of the angles of the triangle.