Question

Let \(p_1, p_2, p_3\) be the altitudes of a triangle ABC drawn through the vertices A, B, C respectively. If \(r_1=4, r_2=6, r_3=12\) are the ex-radii of triangle ABC then \( \frac{1}{p_1^2} + \frac{1}{p_2^2} + \frac{1}{p_3^2} = \)

• A. \( \frac{25}{72} \)
• B. \( \frac{25}{144} \)
• C. \( \frac{25}{288} \)
• D. \( \frac{25}{216} \)

Hint:

For problems involving radii (in-radius, ex-radii) and other triangle properties (altitudes, sides, area), remember these key relations: \( \frac{1}{r} = \sum \frac{1}{r_i} \), \( \Delta^2 = r r_1 r_2 r_3 \), \( \Delta = rs \), and \( r_1 = \Delta/(s-a) \). These allow you to find all properties of the triangle from a few given values.

Answer 1

The Correct Option is C