Solveeit Logo
Login

Question

Question: In any triangle ABC if \(2\cos B = \frac{a}{c}\), then the triangle isC...

In any triangle ABC if 2cos⁡B=ac2\cos B = \frac{a}{c}, then the triangle isC

A

Right angled

B

Equilateral

C

Isosceles

D

None of these

Answer

Isosceles

Explanation

Solution

2cos⁡B=ac=ksin⁡Aksin⁡C=sin⁡Asin⁡C2\cos B = \frac{a}{c} = \frac{k\sin A}{k\sin C} = \frac{\sin A}{\sin C}

⇒ 2cos⁡Bsin⁡C=sin⁡A2\cos B\sin C = \sin A

⇒sin⁡(B+C)−sin⁡(B−C)=sin⁡A\sin(B + C) - \sin(B - C) = \sin A

⇒sin⁡(180o−A)−sin⁡(B−C)\sin(180^{o} - A) - \sin(B - C)

= sinA ⇒ sin⁡A−sin⁡(B−C)=sin⁡A\sin A - \sin(B - C) = \sin A ⇒ sin⁡(B−C)=0\sin(B - C) = 0

⇒ B−C=0B - C = 0 ⇒ B=CB = C

∴\therefore Triangle is isosceles.