Question

 In a \[\Delta ABC\], if \[\dfrac{\cos A}{a}=\dfrac{\cos B}{b}\], show that the triangle is isosceles.

Answer 1

Hint: For the above question, we will have to know about the isosceles triangle. A triangle with two equal sides or it is equivalent to having two equal angles then the triangle is known as an isosceles triangle. Now, we will use the cosine rule of the triangle for angle A and B of the triangle and by using the given condition, we will prove either two sides of the triangle to be equal which shows that the triangle is isosceles.

Complete step by step answer:
We have been given that in \[\Delta ABC\],
\[\dfrac{\cos A}{a}=\dfrac{\cos B}{b}.....\left( i \right)\]
Here, A, B, and C are the angles and a, b and c are the corresponding sides of the triangle as shown in the figure below:

Let us suppose the equation (i) gives a constant ‘k’.
\[\Rightarrow \dfrac{\cos A}{a}=\dfrac{\cos B}{b}=k\]
\[\Rightarrow \cos A=ak\text{ and }\cos B=bk\]
And, the cosine rule in a \[\Delta ABC\], is shown as below:
\[\cos A=\dfrac{{{b}^{2}}+{{c}^{2}}-{{a}^{2}}}{2bc}\]
\[\cos B=\dfrac{{{a}^{2}}+{{c}^{2}}-{{b}^{2}}}{2ac}\]
Now, by substituting the values of cos A = ak and cos B = bk in the cosine rule, we get,
\[ak=\dfrac{{{b}^{2}}+{{c}^{2}}-{{a}^{2}}}{2bc}\]
\[\Rightarrow 2abck={{b}^{2}}+{{c}^{2}}-{{a}^{2}}......\left( ii \right)\]
Also,
\[bk=\dfrac{{{a}^{2}}+{{c}^{2}}-{{b}^{2}}}{2ac}\]
\[\Rightarrow 2abck={{a}^{2}}+{{c}^{2}}-{{b}^{2}}......\left( iii \right)\]
Now, from equation (ii), and (iii), we have observed that the left-hand side of equality have the same value which means the right side will also have the same value.
\[\Rightarrow {{b}^{2}}+{{c}^{2}}-{{a}^{2}}={{a}^{2}}+{{c}^{2}}-{{b}^{2}}\]
On simplifying the above equation, we get,
\[2{{b}^{2}}=2{{a}^{2}}\]
\[\Rightarrow b=a\]
Hence, the sides BC and AC of the given \[\Delta ABC\] are equal.
Therefore, the \[\Delta ABC\] is an isosceles triangle.

Note: In the above type of the question, first draw the diagram and then move further. Also, remember the properties of the triangle as it will help you in these types of questions.