Question

If \( P_1, P_2, P_3 \) are the lengths of the altitudes drawn from the vertices \( A, B, C \) of \( \triangle ABC \) respectively, then: \[ \cos A \cdot \frac{1}{P_1} + \cos B \cdot \frac{1}{P_2} + \cos C \cdot \frac{1}{P_3} = ? \]

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

Hint:

In a triangle, the sum of the cosines of the angles times the reciprocals of the altitudes is related to the circumradius.

Answer 1

The Correct Option is A

The sum of the cosines of the angles times the reciprocals of the altitudes can be related to the circumradius \( R \) using the known formula: \[ \cos A \cdot \frac{1}{P_1} + \cos B \cdot \frac{1}{P_2} + \cos C \cdot \frac{1}{P_3} = \frac{1}{R} \] Thus, the correct answer is option (1) \( \frac{1}{R} \).