Question:
In \( \triangle ABC \), if \( (a - b)(s - c) = (b - c)(s - a) \), then \( r_1, r_2, r_3 \) are in:
In \( \triangle ABC \), if \( (a - b)(s - c) = (b - c)(s - a) \), then \( r_1, r_2, r_3 \) are in:
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In problems involving sides and semiperimeter of a triangle, use algebraic identities to recognize the type of progression formed by related terms.
Updated On: May 13, 2025
- Arithmetic progression
- Geometric progression
- Harmonic progression
- Arithmetic-Geometric progression
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The Correct Option is A
Solution and Explanation
We are given the equation \( (a - b)(s - c) = (b - c)(s - a) \), and we need to determine the type of progression for \( r_1, r_2, r_3 \).
Step 1: The given condition leads to an important identity related to the sides and the semiperimeter of the triangle. This implies that the values \( r_1, r_2, r_3 \) are related in a way that they form an arithmetic progression.
Thus, \( r_1, r_2, r_3 \) are in an arithmetic progression.
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