Circles
Tangent Properties
Grade 11
Question:
<p>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, then the length of DE is:</p>
<p>(a) \(8\sqrt{2}\)</p>
<p>(b) \(16\)</p>
<p>(c) \(9\sqrt{2}\)</p>
<p>(d) \(10\sqrt{2}\)</p>
Step-by-Step Solution
Key Concept: Use the tangency condition (radius perpendicular to tangent) and apply the Pythagorean theorem in the right triangle formed by the center, radius, and tangent point.
<p>Given DC = 12 is the diameter of the large circle, so DC = 12. AC is the diameter of the smaller circle. Since A is the center of the large circle and B is the center of the smaller circle (midpoint of AC), we have AC = 6 (half of DC). The radius of the large circle is 6, and the radius of the small circle is 3. DE is tangent to the small circle at F, so BF ⊥ DE and BF = 3. Using the Pythagorean theorem in the configuration and properties of tangent lines, DE = \(8\sqrt{2}\).</p>
Correct Answer: A