In $\triangle ABC,$ if $2\cos B = \frac{a}{2}$, determine what type of triangle it is.

Question

The given options are (a) right-angled (b) equilateral (c) isosceles

So, I tried using the sine rule, the cosine rule/projection formula, and even the Napier analogy, but I couldn't arrive at a proper answer. I'm guessing it's probably none of the above, but that's unfortunately not one of the given options.

Answer 1

An right angled triangle with hypotenuse equal to 4 units will satisfy the above equation.

Thus we can have an right angled triangle or an isoceles right angled triangle.

So the correct answers are A or A&C

Answer 2

This gives us cos(B)=a/4 . Using cos rule we get

a^2 +c^2 - b^2 / 2ac = a/4 Solving the quadratic for real value of c we get a^2(a^2 - 8)≥ 8b^2 We see that this gives a valid value for a when a=B It can also be right angled with hypotenuse 4 So the answer is A and C also can be equilateral when a=2