Looking at problem 6, I can see a square with an inscribed circle (touching all four sides) and the diagonals of the square drawn. The shaded region is the circle.
Since no dimensions are given for the square, I'll need to work with variables or assume a side length. Let's assume the side length of the square is 's'.
To find the probability that a randomly chosen point falls in the shaded region (the circle), I need to calculate:
Probability = Shaded Area / Total Area
First, let's calculate the total area (square):
Total Area = s × s = s²
Since the circle is inscribed in the square, its diameter equals the side length of the square, which is s. Therefore, the radius is s/2.
Area of the circle = π × r² = π × (s/2)² = π × s²/4 = (πs²)/4
Therefore, the probability is:
Probability = [(πs²)/4] / s² = π/4
This makes sense because the ratio of the area of an inscribed circle to the area of its circumscribing square is always π/4, regardless of the actual size of the square.
Now I have solved all six problems. Let me summarize my answers:
1. Circle divided into 8 sectors with 3 shaded: Probability = 3/8
2. Rectangle with diagonals creating 4 triangles, 2 shaded: Probability = 1/2
3. Rectangle frame (outer 8×6, inner 4×2): Probability = 5/6
4. Square with inscribed circle (side 8): Probability = 1 - π/4
5. Trapezoid (entirely shaded): Probability = 1
6. Square with inscribed circle (no dimensions given): Probability = π/4
These are the solutions to the geometric probability worksheet.
Parent Tip: Review the logic above to help your child master the concept of geometric probability worksheet.
