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:

• A. \( \frac{a}{r_3} \)
• B. \( \frac{b}{r_3} \)
• C. \( \frac{c}{r_3} \)
• D. 1

Hint:

When working with geometric relations in triangles, carefully manipulate the given expressions using algebraic identities to simplify and reach the desired result.

Answer 1

The Correct Option is C

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 \( \frac{c}{r_3} \).