Multiple choice

If $\displaystyle \frac{a}{\sin A}=K,$ then the area of $\displaystyle \Delta ABC$ in terms of K and sines of the angles is -

  1. $\displaystyle \frac{K^{2}}{4}\sin A\sin B\sin C$
  2. $\displaystyle \frac{K^{2}}{2}\sin A\sin B\sin C$
  3. $\displaystyle 2K^{2}\sin A\sin B\sin C\left ( A+B \right )$

  4. none
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

By the Law of Sines, a/sinA = b/sinB = c/sinC = 2R = K. Area = 1/2 * b * c * sinA = 1/2 * (K sinB) * (K sinC) * sinA = K^2/2 * sinA * sinB * sinC.

AI explanation

Using the sine rule, the sides of the triangle can be expressed as a = K sin A, b = K sin B, and c = K sin C. The area of a triangle is given by the formula Area = (1/2)ab sin C. Substituting the expressions for a and b into the area formula yields (1/2)(K sin A)(K sin B) sin C. Simplifying this expression results in (K^2 / 2) sin A sin B sin C.

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