Question:
In the figure shown above, PQRS is a square. The shaded portion is formed by the intersection of sectors of circles with radius equal to the side of the square and centers at S and Q. The probability that any point picked randomly within the square falls in the shaded area is
In the figure shown above, PQRS is a square. The shaded portion is formed by the intersection of sectors of circles with radius equal to the side of the square and centers at S and Q. The probability that any point picked randomly within the square falls in the shaded area is
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To calculate the probability of a point falling in a specific region, divide the area of that region by the total area in question. For geometric problems involving intersections of sectors or other shapes, carefully calculate the area of the overlap.
Updated On: Dec 30, 2025
- \( \frac{4}{\pi} - \frac{\pi}{2} \)
- \( \frac{1}{2} \)
- \( \frac{\pi}{2} - 1 \)
- \( \frac{\pi}{4} \)
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The Correct Option is C
Solution and Explanation
The problem involves finding the probability that a randomly chosen point within the square falls inside the shaded area formed by the intersection of two circular sectors with radius equal to the side length of the square.
Step 1: Understand the geometry of the problem.
- Let the side of the square PQRS be \( r \).
- The shaded area is formed by the intersection of two sectors of circles, each with radius \( r \) and centers at \( S \) and \( Q \).
- The area of each sector is given by: \[ A_{\text{sector}} = \frac{\theta}{360^\circ} \times \pi r^2 \] where \( \theta \) is the central angle of the sector. In this case, \( \theta = 90^\circ \), so the area of each sector is: \[ A_{\text{sector}} = \frac{90^\circ}{360^\circ} \times \pi r^2 = \frac{\pi r^2}{4} \] Step 2: Calculate the area of the shaded region.
- The shaded area is formed by the intersection of two such sectors. The total area of the sectors is \( \frac{\pi r^2}{2} \), but we need to subtract the area of the overlap between the two sectors, which is \( r^2 \). Hence, the area of the shaded region is: \[ A_{\text{shaded}} = \frac{\pi r^2}{2} - r^2 \] Step 3: Calculate the probability.
- The area of the square is \( r^2 \), so the probability that a randomly chosen point falls inside the shaded region is: \[ \text{Probability} = \frac{A_{\text{shaded}}}{A_{\text{square}}} = \frac{\frac{\pi r^2}{2} - r^2}{r^2} = \frac{\pi}{2} - 1 \] Thus, the correct answer is option (C).
Final Answer: \( \frac{\pi}{2} - 1 \)
Step 1: Understand the geometry of the problem.
- Let the side of the square PQRS be \( r \).
- The shaded area is formed by the intersection of two sectors of circles, each with radius \( r \) and centers at \( S \) and \( Q \).
- The area of each sector is given by: \[ A_{\text{sector}} = \frac{\theta}{360^\circ} \times \pi r^2 \] where \( \theta \) is the central angle of the sector. In this case, \( \theta = 90^\circ \), so the area of each sector is: \[ A_{\text{sector}} = \frac{90^\circ}{360^\circ} \times \pi r^2 = \frac{\pi r^2}{4} \] Step 2: Calculate the area of the shaded region.
- The shaded area is formed by the intersection of two such sectors. The total area of the sectors is \( \frac{\pi r^2}{2} \), but we need to subtract the area of the overlap between the two sectors, which is \( r^2 \). Hence, the area of the shaded region is: \[ A_{\text{shaded}} = \frac{\pi r^2}{2} - r^2 \] Step 3: Calculate the probability.
- The area of the square is \( r^2 \), so the probability that a randomly chosen point falls inside the shaded region is: \[ \text{Probability} = \frac{A_{\text{shaded}}}{A_{\text{square}}} = \frac{\frac{\pi r^2}{2} - r^2}{r^2} = \frac{\pi}{2} - 1 \] Thus, the correct answer is option (C).
Final Answer: \( \frac{\pi}{2} - 1 \)
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