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: Understanding the Concept
The altitude from A is \(P_1 = \dfrac{2\Delta}{a}\), and similarly for \(P_2, P_3\). Also \(a = 2R\sin A\).

Step 2: Rewrite the sum
\[ \sum \frac{\cos A}{P_1} = \frac{1}{2\Delta}\sum a\cos A = \frac{R}{2\Delta}\sum \sin 2A \]

Step 3: Evaluate
In a triangle, \(\sin2A + \sin2B + \sin2C = 4\sin A\sin B\sin C\). Also \(\sin A\sin B\sin C = \dfrac{abc}{8R^3} = \dfrac{4R\Delta}{8R^3} = \dfrac{\Delta}{2R^2}\).
\[ \sum \sin 2A = \frac{2\Delta}{R^2} \]
\[ \text{Sum} = \frac{R}{2\Delta}\cdot\frac{2\Delta}{R^2} = \frac{1}{R} \]

Final Answer:
The value is \(\frac1R\), option (B). \[ \boxed{\frac{1}{R}} \]