Question

With usual notations, if in a triangle ABC,$\dfrac{{b + c}}{{11}} = \dfrac{{c + a}}{{12}} = \dfrac{{a + b}}{{13}}$ then $\cos A:\cos B:\cos C$is equal to
A) $\left( a \right){\text{ 7:19:25}}$
B) $\left( b \right){\text{ 19:7:25}}$
C) $\left( c \right){\text{ 12:14:20}}$
D) $\left( d \right){\text{ 19:25:20}}$

Answer 1

For solving this type of question, we will assume the ratio be any arbitrary constant and from there we will calculate the sides of a triangle. And after that putting the values in the ratios we have to calculate, we will get the required ratios.

Complete step by step solution:
Let us assume each ratio be$k$, then we have the equation written as
$\Rightarrow b + c = 11k,{\text{ }}c + a = 12k,{\text{ }}a + b = 13k$
Now from these three equations, on adding all the three equations, we will get
$\Rightarrow 2\left( {a + b + c} \right) = 36k$
Taking the $2$to the right side, we get
$\Rightarrow a + b + c = 18k$
Now using the above equation, after calculating the values we get
$\Rightarrow a = 7k,{\text{ b = 6k, c = 5k}}$
So we have all the values of the term.
Now we will calculate the values of $\cos A:\cos B:\cos C$
So the values $\cos A$will be equals to
$\Rightarrow \dfrac{{{b^2} + {c^2} - {a^2}}}{{2bc}}$
On putting the values, we get
$\Rightarrow \dfrac{{36{k^2} + 25{k^2} - 49{k^2}}}{{60{k^2}}}$
And on calculating, we get
$\Rightarrow \dfrac{{12{k^2}}}{{60{k^2}}}$
And on doing the lowest common factor, we get
$\Rightarrow \cos A = \dfrac{1}{5}$
So the values $\cos B$will be equals to
$\Rightarrow \dfrac{{{a^2} + {c^2} - {b^2}}}{{2ac}}$
On putting the values, we get
$\Rightarrow \dfrac{{49{k^2} + 25{k^2} - 36{k^2}}}{{70{k^2}}}$
And on calculating, we get
$\Rightarrow \dfrac{{38{k^2}}}{{70{k^2}}}$
And on doing the lowest common factor, we get
$\Rightarrow \cos B = \dfrac{{19}}{{35}}$
So the values $\cos C$will be equals to
$\Rightarrow \dfrac{{{a^2} + {b^2} - {c^2}}}{{2ab}}$
On putting the values, we get
$\Rightarrow \dfrac{{49{k^2} + 36{k^2} - 25{k^2}}}{{84{k^2}}}$
And on calculating, we get
$\Rightarrow \dfrac{{60{k^2}}}{{84{k^2}}}$
And on doing the lowest common factor, we get
$\Rightarrow \cos A = \dfrac{5}{7}$
Therefore, the ratios $\cos A:\cos B:\cos C$will be
$\Rightarrow \dfrac{1}{5}:\dfrac{{19}}{{35}}:\dfrac{5}{7}$
So it can also be written as
$\Rightarrow 7:19:25$

Hence, the option $\left( a \right)$will be correct.

Note:
For solving this type of question, we should always assume the ratios be any constant and then go for the solution of it. In this type of question if our linear equations are correct then we can easily find out the ratios. And in this way, we can easily solve it.