The figure above shows a semicircle with center O and a quarter circle with center R. If OQ = 4 and QR = 6, what is the ratio of the area of the shaded region to the area of the semicircular region?
Show Hint
In ratio problems involving geometric areas with \(\pi\), the \(\pi\) term will often cancel out. Focus on correctly identifying the radii and using the correct fractions for the parts of the circle (1/2 for a semicircle, 1/4 for a quarter circle).
Step 1: Understanding the Concept:
This is a geometry problem involving the areas of parts of circles. We need to find the area of a semicircle and the area of a shaded quarter circle, and then find the ratio between them. Step 2: Key Formula or Approach:
The area of a full circle is given by \( A = \pi r^2 \), where \(r\) is the radius.
- Area of a semicircle = \( \frac{1}{2} \pi r^2 \)
- Area of a quarter circle = \( \frac{1}{4} \pi r^2 \) Step 3: Detailed Explanation: 1. Area of the Semicircular Region:
The semicircle has its center at O. The segment OQ is a radius of this semicircle.
We are given OQ = 4. So, the radius of the semicircle is \( r_{\text{semi}} = 4 \).
\[ \text{Area}_{\text{semicircle}} = \frac{1}{2} \pi (r_{\text{semi}})^2 = \frac{1}{2} \pi (4)^2 = \frac{1}{2} \pi (16) = 8\pi \]
2. Area of the Shaded Region:
The shaded region is a quarter circle with its center at R.
We need to find the radius of this quarter circle. From the diagram, the segment QR is the radius of the quarter circle.
We are given QR = 6. So, the radius of the quarter circle is \( r_{\text{quarter}} = 6 \).
\[ \text{Area}_{\text{shaded}} = \text{Area}_{\text{quarter circle}} = \frac{1}{4} \pi (r_{\text{quarter}})^2 = \frac{1}{4} \pi (6)^2 = \frac{1}{4} \pi (36) = 9\pi \]
3. Ratio of the Areas:
The question asks for the ratio of the area of the shaded region to the area of the semicircular region.
\[ \text{Ratio} = \frac{\text{Area}_{\text{shaded}}}{\text{Area}_{\text{semicircle}}} = \frac{9\pi}{8\pi} \]
The \( \pi \) terms cancel out.
\[ \text{Ratio} = \frac{9}{8} \]
In ratio notation, this is 9:8. Step 4: Final Answer:
The ratio of the area of the shaded region to the area of the semicircular region is 9:8.
