Question

In any triangle $ABC$, the value of $a({{b}^{2}}+{{c}^{2}})\cos A+b({{c}^{2}}+{{a}^{2}})\cos B+c({{a}^{2}}+{{b}^{2}})\cos C$ is
A. $3ab{{c}^{2}}$
B. $3{{a}^{2}}bc$
C. $3abc$
D. $3a{{b}^{2}}c$

Answer 1

Hint: To solve this question, we will use all the three projection formulas. We will first simplify the given equation and rewrite it. We will then apply the projection formulas and determine the value of the equation.
Projection formula can be defined as the length of any side of a triangle is equal to the sum of the projections of two other sides on it.

Formula Used: There are three projection formulas:
\[\begin{align}
  & a=c\cos B+b\cos C \\
 & b=c\cos A+a\cos C \\
 & c=b\cos A+a\cos B \\
\end{align}\]

Complete step by step solution: We are given a triangle $ABC$ and we have to determine the value of $a({{b}^{2}}+{{c}^{2}})\cos A+b({{c}^{2}}+{{a}^{2}})\cos B+c({{a}^{2}}+{{b}^{2}})\cos C$.
We will simplify the equation $a({{b}^{2}}+{{c}^{2}})\cos A+b({{c}^{2}}+{{a}^{2}})\cos B+c({{a}^{2}}+{{b}^{2}})\cos C$ and derive its value.
$\begin{align}
  & =(a{{b}^{2}}+a{{c}^{2}})\cos A+(b{{c}^{2}}+b{{a}^{2}})\cos B+(c{{a}^{2}}+c{{b}^{2}})\cos C \\
 & =a{{b}^{2}}\cos A+a{{c}^{2}}\cos A+b{{c}^{2}}\cos B+b{{a}^{2}}\cos B+c{{a}^{2}}\cos C+c{{b}^{2}}\cos C \\
\end{align}$
We will now rewrite the equation.
$\begin{align}
  & =a{{b}^{2}}\cos A+b{{a}^{2}}\cos B+a{{c}^{2}}\cos A+c{{a}^{2}}\cos C+b{{c}^{2}}\cos B+c{{b}^{2}}\cos C \\
 & =ab(b\cos A+a\cos B)+ac(c\cos A+a\cos C)+bc(c\cos B+b\cos C) \\
\end{align}$
We will now substitute the projection formula in the above equation.
$\begin{align}
  & =abc+abc+abc \\
 & =3abc \\
\end{align}$
The value of $a({{b}^{2}}+{{c}^{2}})\cos A+b({{c}^{2}}+{{a}^{2}})\cos B+c({{a}^{2}}+{{b}^{2}})\cos C$ for the triangle $ABC$ is $3abc$. Hence the correct option is (C).

Note: There are three types of a triangle in terms of angle. It can be either acute, obtuse or right angled triangle. In terms of the shape there are also three kind of triangle scalene, isosceles and equilateral triangle. Some properties of triangles apply equally on every triangle however some properties are specific to a type of triangle.
For example- To calculate an area of a triangle the Heron’s formula applies on every triangle but the area of the right angled triangle is specific to it only. The angle sum property applies on every triangle irrespective of their type on the basis of angle or shape.
Hence we should once check before using any properties of a triangle that if it is applicable or not.