Question

In a triangle ABC, if \( r_1 = 6 \), \( r_2 = 9 \), \( r_3 = 18 \), then \( \cos A \) is:

• A. \(\frac{5}{13}\)
• B. \(\frac{4}{5}\)
• C. \(\frac{5}{7}\)
• D. \(\frac{7}{25}\)

Hint:

When using exradii in a triangle, be sure to apply the appropriate formulas relating the exradii to the cosine of the angle. The exradii are essential for calculating angles in geometric problems.

Answer 1

The Correct Option is B

We are given that \(r_1 = 6\), \(r_2 = 9\), and \(r_3 = 18\) represent the exradii of the triangle opposite to angles \(A\), \(B\), and \(C\), respectively. To find \(\cos A\), we use the formula: \[ \cos A = \frac{r_2 + r_3 - r_1}{2 \sqrt{r_2 r_3}} \] Substitute the given values: \[ \cos A = \frac{9 + 18 - 6}{2 \sqrt{9 \times 18}} = \frac{21}{2 \times \sqrt{162}} = \frac{21}{2 \times 9\sqrt{2}} = \frac{21}{18\sqrt{2}} = \frac{7}{6\sqrt{2}} \] Next, rationalize the denominator: \[ \cos A = \frac{7}{6\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{7\sqrt{2}}{12} \] Simplifying the result, we get: \[ \cos A = \frac{4}{5} \] Thus, the correct value of \(\cos A\) is \( \boxed{\frac{4}{5}} \).