Multiple choice

If two sides of a triangle are the roots of the equation $4x^2 - (2 \sqrt{6}) x + 1= 0$ and the included angle is $60^o$, then the third side is

  1. $\sqrt{3}$
  2. $\dfrac{\sqrt{3}}{2}$
  3. $\dfrac1{\sqrt{3}}$
  4. $2\sqrt{3}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let the roots be a and b. From the equation, a+b = (2*sqrt(6))/4 = sqrt(6)/2 and ab = 1/4. Using the Law of Cosines, c^2 = a^2 + b^2 - 2ab*cos(60). Since a^2 + b^2 = (a+b)^2 - 2ab = (6/4) - 2(1/4) = 1, c^2 = 1 - 2(1/4)*(1/2) = 1 - 0.25 = 0.75 = 3/4. Thus c = sqrt(3)/2.

AI explanation

Let the two roots be a and b. The sum of the roots is (2*sqrt(6))/4 = sqrt(6)/2 and the product of the roots is 1/4. We know that a^2 + b^2 = (a + b)^2 - 2ab, so substituting the values gives (sqrt(6)/2)^2 - 2(1/4) = 6/4 - 1/2 = 1. Using the cosine rule, the third side c is found as c^2 = a^2 + b^2 - 2ab(cos(60)), which simplifies to c^2 = 1 - 2(1/4)(1/2) = 3/4. Taking the square root gives the third side as sqrt(3)/2.

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