Question

In \( \triangle ABC \), evaluate: \[ \frac{r_2 (r_1 + r_3)}{\sqrt{r_1 r_2 + r_2 r_3 + r_3 r_1}}. \]

• A. \( a \)
• B. \( b \)
• C. \( c \)
• D. \( s \)

Hint:

For inradius and semiperimeter problems, express values in terms of area and side lengths for simplifications.

Answer 1

The Correct Option is B

To evaluate the expression: \[ \frac{r_2 (r_1 + r_3)}{\sqrt{r_1 r_2 + r_2 r_3 + r_3 r_1}}, \] we first recall the definitions of \( r_1, r_2, r_3 \) in \( \triangle ABC \): - \( r_1 = \frac{\Delta}{s - a} \), - \( r_2 = \frac{\Delta}{s - b} \), - \( r_3 = \frac{\Delta}{s - c} \), where \( \Delta \) is the area of the triangle, and \( s = \frac{a + b + c}{2} \) is the semi-perimeter. Step 1: Express \( r_1, r_2, r_3 \) in terms of \( \Delta \) and \( s \) Substitute the expressions for \( r_1, r_2, r_3 \): \[ r_1 = \frac{\Delta}{s - a}, \quad r_2 = \frac{\Delta}{s - b}, \quad r_3 = \frac{\Delta}{s - c}. \] Step 2: Simplify the numerator The numerator is: \[ r_2 (r_1 + r_3) = \frac{\Delta}{s - b} \left( \frac{\Delta}{s - a} + \frac{\Delta}{s - c} \right). \] Factor out \( \Delta \): \[ r_2 (r_1 + r_3) = \frac{\Delta^2}{s - b} \left( \frac{1}{s - a} + \frac{1}{s - c} \right). \] Combine the fractions: \[ r_2 (r_1 + r_3) = \frac{\Delta^2}{s - b} \cdot \frac{(s - c) + (s - a)}{(s - a)(s - c)}. \] Simplify the numerator: \[ (s - c) + (s - a) = 2s - (a + c) = b. \] Thus: \[ r_2 (r_1 + r_3) = \frac{\Delta^2}{s - b} \cdot \frac{b}{(s - a)(s - c)}. \] Step 3: Simplify the denominator The denominator is: \[ \sqrt{r_1 r_2 + r_2 r_3 + r_3 r_1}. \] Substitute the expressions for \( r_1, r_2, r_3 \): \[ r_1 r_2 + r_2 r_3 + r_3 r_1 = \frac{\Delta^2}{(s - a)(s - b)} + \frac{\Delta^2}{(s - b)(s - c)} + \frac{\Delta^2}{(s - c)(s - a)}. \] Factor out \( \Delta^2 \): \[ r_1 r_2 + r_2 r_3 + r_3 r_1 = \Delta^2 \left( \frac{1}{(s - a)(s - b)} + \frac{1}{(s - b)(s - c)} + \frac{1}{(s - c)(s - a)} \right). \] Combine the fractions: \[ r_1 r_2 + r_2 r_3 + r_3 r_1 = \Delta^2 \cdot \frac{(s - c) + (s - a) + (s - b)}{(s - a)(s - b)(s - c)}. \] Simplify the numerator: \[ (s - c) + (s - a) + (s - b) = 3s - (a + b + c) = 3s - 2s = s. \] Thus: \[ r_1 r_2 + r_2 r_3 + r_3 r_1 = \Delta^2 \cdot \frac{s}{(s - a)(s - b)(s - c)}. \] Take the square root: \[ \sqrt{r_1 r_2 + r_2 r_3 + r_3 r_1} = \Delta \cdot \sqrt{\frac{s}{(s - a)(s - b)(s - c)}}. \] Step 4: Compute the expression The expression becomes: \[ \frac{r_2 (r_1 + r_3)}{\sqrt{r_1 r_2 + r_2 r_3 + r_3 r_1}} = \frac{\frac{\Delta^2 b}{(s - a)(s - b)(s - c)}}{\Delta \cdot \sqrt{\frac{s}{(s - a)(s - b)(s - c)}}}. \] Simplify: \[ \frac{\Delta^2 b}{(s - a)(s - b)(s - c)} \cdot \frac{1}{\Delta \cdot \sqrt{\frac{s}{(s - a)(s - b)(s - c)}}} = \frac{\Delta b}{\sqrt{s (s - a)(s - b)(s - c)}}. \] Recall that \( \Delta = \sqrt{s(s - a)(s - b)(s - c)} \), so: \[ \frac{\Delta b}{\Delta} = b. \] Final Answer: \[ \boxed{b} \]