Geometric Probability Worksheet with Diagrams for Calculating Probabilities

Here are the step-by-step solutions for the geometric probability problems on the worksheet.

1. Rectangle with a Triangle
* Total Area: The rectangle has a width of 8 and a height of 6. Area = $8 \times 6 = 48$.
* Shaded Area: The shaded part is a triangle with base 8 and height 6. Area = $\frac{1}{2} \times 8 \times 6 = 24$.
* Probability: $\frac{24}{48}$ simplifies to $\frac{1}{2}$.

2. Circle Sector
* Total Angle: A full circle is $360^\circ$.
* Shaded Angle: The shaded sector is $90^\circ$.
* Probability: $\frac{90}{360}$ simplifies to $\frac{1}{4}$.

3. Octagon
* Total Parts: The octagon is divided into 8 equal triangles.
* Shaded Parts: 5 of the triangles are shaded.
* Probability: $\frac{5}{8}$.

4. Large Triangle
* Total Parts: The large triangle is divided into 16 small, equal triangles.
* Shaded Parts: Counting the shaded ones, there are 6.
* Probability: $\frac{6}{16}$ simplifies to $\frac{3}{8}$.

5. Concentric Circles (Ring)
* Inner Radius ($r$): 2. Area = $\pi(2^2) = 4\pi$.
* Outer Radius ($R$): 7. Total Area = $\pi(7^2) = 49\pi$.
* Shaded Area (Ring): Subtract the inner area from the total area: $49\pi - 4\pi = 45\pi$.
* Probability: $\frac{45\pi}{49\pi}$ simplifies to $\frac{45}{49}$.

6. Trapezoid
* Total Area: Use the formula $\frac{1}{2}(b_1 + b_2)h$. Bases are 4 and 10, height is 4. Area = $\frac{1}{2}(14)(4) = 28$.
* Shaded Area: This is a triangle with base 10 and height 4. Area = $\frac{1}{2}(10)(4) = 20$.
* Probability: $\frac{20}{28}$ simplifies to $\frac{5}{7}$.

7. Rectangle with Two Circles
* Circle Dimensions: Diameter is 4, so radius is 2.
* Rectangle Dimensions: Height is two diameters ($4+4=8$). Width is one diameter (4). Total Area = $8 \times 4 = 32$.
* Shaded Area: Two circles. Area of one = $\pi(2^2) = 4\pi$. Two circles = $8\pi$.
* Probability: $\frac{8\pi}{32}$ simplifies to $\frac{\pi}{4}$.

8. Rectangle with Diagonals
* Geometry Rule: When diagonals cross in a rectangle, they create 4 triangles of equal area.
* Shaded Parts: 1 out of the 4 triangles is shaded.
* Probability: $\frac{1}{4}$.

9. Square Inscribed in a Circle
* Circle Radius: Diameter is 12, so radius is 6. Total Area = $\pi(6^2) = 36\pi$.
* Square Area: The diagonal of the square equals the diameter of the circle (12). Area of a square using diagonal $d$ is $\frac{d^2}{2}$. Area = $\frac{12^2}{2} = \frac{144}{2} = 72$.
* Probability: $\frac{72}{36\pi}$ simplifies to $\frac{2}{\pi}$.

10. Diamond (Rhombus)
* Total Parts: The shape is divided into 8 small, equal right triangles.
* Shaded Parts: 2 of the triangles are shaded.
* Probability: $\frac{2}{8}$ simplifies to $\frac{1}{4}$.

11. Rectangle with Shapes
* Dimensions: Width = $4 + 4 = 8$. Height = $4 + 4 = 8$. It is an $8 \times 8$ square. Total Area = 64.
* Shaded Parts:
* Top-left square: $4 \times 4 = 16$.
* Bottom-right square: $4 \times 4 = 16$.
* Bottom-left region: A $4 \times 4$ square minus a circle (radius 2). Area = $16 - 4\pi$.
* Total Shaded Area = $16 + 16 + (16 - 4\pi) = 48 - 4\pi$.
* Probability: $\frac{48 - 4\pi}{64}$. Divide top and bottom by 16 to get $\frac{3 - \frac{\pi}{4}}{4}$ or $\frac{12 - \pi}{16}$.

12. Rectangle with Semicircles
* Dimensions: Width = 4. Height = $2 + 2 + 2 = 6$. Total Area = $4 \times 6 = 24$.
* Unshaded Parts (White): There are 2 white semicircles with radius 2. Together they make 1 full circle. Area = $\pi(2^2) = 4\pi$.
* Shaded Area: Total Area - Unshaded Area = $24 - 4\pi$.
* Probability: $\frac{24 - 4\pi}{24}$. Simplify by dividing by 4: $\frac{6 - \pi}{6}$.

Final Answer:
1. 1/2
2. 1/4
3. 5/8
4. 3/8
5. 45/49
6. 5/7
7. π/4
8. 1/4
9. 2/π
10. 1/4
11. (12 - π)/16
12. (6 - π)/6