Question

In \( \triangle ABC \), if \( \frac{1}{r_1}, \frac{1}{r_2}, \frac{1}{r_3} \) are in arithmetic progression, then \( r_1 : r_2 : r_3 = \)?

• A. \( 3 : 2 \)
• B. \( 2 : 1 \)
• C. \( 1 : 3 \)
• D. \( 3 : 1 \)

Hint:

When the reciprocals of terms are in arithmetic progression, the terms themselves follow a specific ratio that can be found by applying the properties of arithmetic progression.

Answer 1

The Correct Option is D

Given that \( \frac{1}{r_1}, \frac{1}{r_2}, \frac{1}{r_3} \) are in arithmetic progression, we can use the property of arithmetic progression. For any three terms in arithmetic progression, the middle term is the average of the other two. Therefore, we have the relation: \[ \frac{1}{r_2} = \frac{1}{2} \left( \frac{1}{r_1} + \frac{1}{r_3} \right) \] This implies that the ratio of the radii is \( r_1 : r_2 : r_3 = 3 : 1 \). Thus, the correct answer is option (4) \( 3 : 1 \).