Trigonometry & Inverse Trigonometry
Trigonometry
star_batch_jee_advanced_2025
Grade 11
Question:
In $\triangle ABC$, if circumradius $'R'$ and inradius $'r'$ are connected by relation $R^2 - 4Rr + 8r^2 - 12r + 9 = 0$, then the greatest integer which is less than the semiperimeter of $\triangle ABC$ is:
Step-by-Step Solution
Key Concept: Recognize the Euler inequality constraint $(R - 2r)^2 \geq 0$ becomes an equality only for equilateral triangles.
Expanding $(R^2 - 4Rr + 4r^2) + (4r^2 - 12r + 9) = 0$ gives $(R - 2r)^2 + (2r - 3)^2 = 0$. This implies $R = 2r$ and $r = \frac{3}{2}$, so $R = 3$. For a triangle with this relationship, $\triangle ABC$ must be equilateral.
Correct Answer: 7