Question:
Let C be a circle with center P, and AB be a diameter of C. Suppose P1 is the midpoint of the line segment PB, P2 is the midpoint of the line segment P1B, and so on. Let C1, C2, C3, C4, ... be circles with diameters P1P2, P2P3, P3P4, ... respectively. Suppose the circles C1, C2, C3, ... are all shaded. The ratio of the area of the unshaded portion of C to that of the original circle is
Let C be a circle with center P, and AB be a diameter of C. Suppose P1 is the midpoint of the line segment PB, P2 is the midpoint of the line segment P1B, and so on. Let C1, C2, C3, C4, ... be circles with diameters P1P2, P2P3, P3P4, ... respectively. Suppose the circles C1, C2, C3, ... are all shaded. The ratio of the area of the unshaded portion of C to that of the original circle is
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When dealing with geometric shapes with progressive scaling, use the properties of similar figures and geometric progressions.
Updated On: Aug 1, 2025
- 8 : 9
- 9 : 10
- 10 : 11
- 11 : 12
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The Correct Option is B
Solution and Explanation
The ratio of the area of the unshaded portion to the original circle can be found by using the geometric series, as the diameters of the circles reduce in a pattern. Based on the construction and the ratio of areas, the required ratio is \( 9 : 10 \).
\[
\boxed{9 : 10}
\]
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