Question

Let \(P_1\), \(P_2\), \(P_3\) be the altitudes of a triangle ABC from the vertices A, B, C respectively. If \(△\) denotes the area of the triangle and s is the semi-perimeter of the triangle, then \(\frac{cosA}{P_1}+\frac{cosB}{P_2}+\frac{cosC}{P_3} =\)

• A. \(R\)
• B. \(\frac{1}{R}\)
• C. \(R^2\)
• D. \(\frac{1}{R^2}\)

Hint:

Write each altitude as 2*Area/side and use a = 2R sin A.

Answer 1

The Correct Option is B

Step 1: Another form of the altitude
The altitude from A is $P_1 = b\sin C = 2R\sin B\sin C$, and likewise for the others. So $\dfrac{\cos A}{P_1} = \dfrac{\cos A\sin A}{2R\sin A\sin B\sin C}$.

Step 2: Add the three terms
The sum is $\dfrac{\sin A\cos A + \sin B\cos B + \sin C\cos C}{2R\sin A\sin B\sin C} = \dfrac{\frac12(\sin 2A + \sin 2B + \sin 2C)}{2R\sin A\sin B\sin C}$.

Step 3: Use the identity
Since $\sin 2A + \sin 2B + \sin 2C = 4\sin A\sin B\sin C$, the sum is $\dfrac{2\sin A\sin B\sin C}{2R\sin A\sin B\sin C} = \dfrac1R$. Check with an equilateral triangle of side 1: the sum is $\sqrt3$ and $R = 1/\sqrt3$, so $1/R = \sqrt3$. Option (B).

Final Answer:
Option (B). \[ \boxed{\frac{1}{R}} \]