Circles
Circle
Allen Star Batch
Grade 11
Question:
A circle is inscribed in an equilateral triangle with side lengths 6 units. Another circle is drawn inside the triangle (but outside the first circle), tangent to the first circle and two of the sides of the triangle. The radius of the smaller circle is:
$1/\sqrt{3}$
$2/3$
$1/2$
1
Step-by-Step Solution
Key Concept: Use the angle condition $\sin 30° = \frac{1}{2}$ to relate the inradius $r$ to the geometric configuration.
Given $R + r + x = GB$ where $x = PB$ and $\sqrt{3} + r + x = 2\sqrt{3}$, we find $R = 3\sqrt{3} - \frac{1}{3} - \sqrt{3}$. Since $x = \sqrt{3} - r = PB$ and $\sin 30° = \frac{r}{\sqrt{3} - r} = \frac{1}{2}$, solving gives $r = \frac{1}{\sqrt{3}}$.
Correct Answer: 1