Question

In a triangle ABC, if $3\angle A = 4\angle B = 6\angle C$, then calculate the angles.

Answer 1

Hint: In the given question, we are provided with a relation between three angles of a triangle. So, we have to first calculate the ratio of the three angles of the triangle by dividing all the sides of the given relation between the three angles by a suitable number. Then, we know that the sum of the three angles of a triangle is ${180^ \circ }$. So, we first assume the angles of a triangle in terms of a single variable with the help of the ratio available to us and then add up these angles employing the angle sum property of the triangle.

Complete step by step answer:
So, we are given relations in the three angles of triangle ABC as $3\angle A = 4\angle B = 6\angle C$. So, we will divide all the sides of this relation by a suitable number so as to obtain the ratio of these angles.The suitable number is the LCM of $3$, $4$ and $6$ so that there is no number left in the numerator.We know that the LCM of $3$, $4$ and $6$ is $12$.

Hence, dividing all sides of relation by $12$, we get,
$ \Rightarrow \dfrac{{3\angle A}}{{12}} = \dfrac{{4\angle B}}{{12}} = \dfrac{{6\angle C}}{{12}}$
Cancelling the common factors in numerator and denominator, we get,
$ \Rightarrow \dfrac{{\angle A}}{4} = \dfrac{{\angle B}}{3} = \dfrac{{\angle C}}{2}$
Now, let all the three sides of the relation be equal to x.
Then, we get, $\dfrac{{\angle A}}{4} = \dfrac{{\angle B}}{3} = \dfrac{{\angle C}}{2} = x$.
So, the angle A of the triangle ABC $ = 4x$
Angle B of the triangle ABC $ = 3x$
Similarly, the angle C $ = 2x$

This is also in accordance with the relation between the three angles provided to us in the question itself. Also, we know that the sum of three angles of the triangle is ${180^ \circ }$.So, we get,
$ \Rightarrow \angle A + \angle B + \angle C = {180^ \circ }$
Substituting the values of the angles in terms of x, we get,
$ \Rightarrow 4x + 3x + 2x = {180^ \circ }$
$ \Rightarrow 9x = {180^ \circ }$
Dividing both the sides of equation by $9$, we get,
$ \Rightarrow x = {20^ \circ }$
So, we get the value of variable x as ${20^ \circ }$.
Hence, the angle A \[ = 4x = {80^ \circ }\]
Angle B $ = 3x = {60^ \circ }$
Angle C $ = 2x = {40^ \circ }$

Hence, the value of angles A,B and C are $80^ \circ,60^ \circ$ and $40^ \circ$.

Note:We must know the method of finding the ratio of the three angles from the relation between the three angles of the triangle. There is no fixed way of solving a given algebraic equation. Algebraic equations can be solved in various ways. Method of transposition involves doing the exact same thing on both sides of an equation with the aim of bringing like terms together and isolating the variable or the unknown term in order to simplify the equation and finding the value of the required parameter.