Circles
Circle
Allen Star Batch
Grade 11
Question:
In the diagram, $DC$ is a diameter of the large circle centered at $A$, and $AC$ is a diameter of the smaller circle centered at $B$. If $DE$ is tangent to the smaller circle at $F$ and $DC = 12$ units then the length of $DE$ is:
$13$
$16$
$8\sqrt{2}$
$10\sqrt{2}$
Step-by-Step Solution
Key Concept: Chord length formula $2r\cos\theta$ relates the geometric angle at the center to the actual chord distance.
Given $\sin\theta = \frac{r}{3r} = \frac{1}{3}$ where $\angle EDC = \theta$, we have $\cos\theta = \frac{2\sqrt{2}}{3}$. The chord length is $DE = 2(2r\cos\theta) = 4 \times \frac{2\sqrt{2}}{3} = 8\sqrt{2}$.
Correct Answer: 3