Question

The sides of a triangle are 5, 12 and 13. Find the area of the triangle.

Answer 1

Hint: Here, we will first find the semi perimeter of the triangle, that is, \[s = \dfrac{{a + b + c}}{2}\]. Then use the heron’s formula of area of the triangle is \[Area = \sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \], where \[s\] is the semi perimeter of the triangle and \[a\], \[b\] and \[c\] are the sides of the triangle. Apply this formula to the area of the triangle, and then use the given sides to find the required value.

Complete step-by-step answer:
We are given the sides of a triangle are 5, 12 and 13.
We know that the semi perimeter of the circle is calculated by dividing the sum of three sides by 2, that is, \[s = \dfrac{{a + b + c}}{2}\].
We will find the values of \[a\], \[b\] and \[c\] are the sides of the triangle.
\[a = 5\]
\[b = 12\]
\[c = 13\]
We will now substitute these values in the above formula to find the semi perimeter of the given triangle.
\[
   \Rightarrow s = \dfrac{{5 + 12 + 13}}{2} \\
   \Rightarrow s = \dfrac{{30}}{2} \\
   \Rightarrow s = 15 \\
 \]
We know that the area of the triangle is calculated by\[Area = \sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \], where \[s\] is the semi perimeter of the triangle and \[a\], \[b\] and \[c\] are the three sides of the triangle.
Substituting these values of the semi perimeter of the triangle and the sides of the triangle \[a\], \[b\] and \[c\] in the above formula of area of the triangle, we get
\[
  Area = \sqrt {15\left( {15 - 5} \right)\left( {15 - 12} \right)\left( {15 - 13} \right)} \\
   = \sqrt {15 \times 10 \times 3 \times 2} \\
   = \sqrt {900} \\
   = 30 \\
 \]
Thus, the area of the triangle is \[30 \].

Note: In solving these types of questions, Heron’s formula should be used to compute the area of a triangle where sides of the triangle are given. Students can also find the area of triangle of using the formula, \[{\text{Area}} = \dfrac{1}{2} \times {\text{base}} \times {\text{height}}\], but this way could take a little longer as we have to find the height of the triangle first.