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:
\[
\cos A \cdot \frac{1}{P_1} + \cos B \cdot \frac{1}{P_2} + \cos C \cdot \frac{1}{P_3} = ?
\]
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:
\[
\cos A \cdot \frac{1}{P_1} + \cos B \cdot \frac{1}{P_2} + \cos C \cdot \frac{1}{P_3} = ?
\]
Show Hint
In a triangle, the sum of the cosines of the angles times the reciprocals of the altitudes is related to the circumradius.
Updated On: May 15, 2025
- \( \frac{1}{R} \)
- \( R \)
- \( \frac{\Delta}{R} \)
- \( \frac{1}{R} \)
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The Correct Option is A
Solution and Explanation
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) \( \frac{1}{R} \).
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