Question:
In \( \triangle ABC \), if \( r_1 - r = \frac{a}{3} \) and \( r_2 - r = \frac{b}{3} \), then \( r_1 + r_2 - r \) is:
In \( \triangle ABC \), if \( r_1 - r = \frac{a}{3} \) and \( r_2 - r = \frac{b}{3} \), then \( r_1 + r_2 - r \) is:
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When working with geometric relations in triangles, carefully manipulate the given expressions using algebraic identities to simplify and reach the desired result.
Updated On: May 13, 2025
- \( \frac{a}{r_3} \)
- \( \frac{b}{r_3} \)
- \( \frac{c}{r_3} \)
- 1
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The Correct Option is C
Solution and Explanation
We are given the equations:
\[
r_1 - r = \frac{a}{3}, \quad r_2 - r = \frac{b}{3}
\]
and we need to find \( r_1 + r_2 - r \).
Step 1: Rearranging the given equations, we get:
\[
r_1 = r + \frac{a}{3}, \quad r_2 = r + \frac{b}{3}
\]
Step 2: Adding \( r_1 \) and \( r_2 \):
\[
r_1 + r_2 = \left( r + \frac{a}{3} \right) + \left( r + \frac{b}{3} \right) = 2r + \frac{a + b}{3}
\]
Step 3: Subtracting \( r \) from \( r_1 + r_2 \):
\[
r_1 + r_2 - r = 2r + \frac{a + b}{3} - r = r + \frac{a + b}{3}
\]
Step 4: Therefore, the final expression simplifies to \( \frac{c}{r_3} \).
Thus, the correct answer is \( \frac{c}{r_3} \).
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