Geometric Probability - Free Printable

I will solve the geometric probability problems shown in the image. The task is to find the probability that a randomly chosen point in each figure lies in the shaded region. This is calculated by dividing the area of the shaded region by the total area of the figure.

I'll solve each problem step by step, starting with problem 1.

Problem 1: Right Triangle

First, look closely: The figure is a right triangle with legs of length 12 and 8. The shaded region is a smaller right triangle within it, sharing the same right angle, with legs of length 6 and 4. I need to calculate the areas of both triangles.

Next, find information: I can calculate this directly using the formula for the area of a triangle: Area = (1/2) * base * height.

- Area of the large triangle = (1/2) * 12 * 8 = 48
- Area of the small shaded triangle = (1/2) * 6 * 4 = 12
- Probability = Shaded Area / Total Area = 12 / 48 = 1/4 = 25%

The answer for problem 1 is 25%.

---

Problem 2: Trapezoid

First, look closely: The figure is a trapezoid with an outer dimension of 20 units on top and 8 units on the bottom, and a height of 8 units. Inside it is a smaller, unshaded trapezoid with a top base of 12 and a bottom base of 8, also with a height of 8. The shaded region is the area between the two trapezoids.

Next, find information: I need to calculate the area of the large trapezoid and subtract the area of the small trapezoid to get the shaded area.

- Area of a trapezoid = (1/2) * (sum of parallel bases) * height
- Area of large trapezoid = (1/2) * (20 + 8) * 8 = (1/2) * 28 * 8 = 112
- Area of small trapezoid = (1/2) * (12 + 8) * 8 = (1/2) * 20 * 8 = 80
- Shaded Area = 112 - 80 = 32
- Probability = Shaded Area / Total Area = 32 / 112 = 2/7 ≈ 28.57%

The answer for problem 2 is approximately 28.57%, which rounds to 29% if we are choosing from the given options.

---

Problem 3: Circle with Sectors

First, look closely: The figure is a circle with a radius of 8 inches. There are two shaded sectors. One sector has a central angle of 120 degrees, and the other has a central angle of 60 degrees. The total shaded area is the sum of these two sectors.

Next, find information: I need to calculate the area of the entire circle and the combined area of the two shaded sectors.

- Area of a circle = π * r² = π * 8² = 64π
- Area of a sector = (θ/360) * π * r²
- Area of 120-degree sector = (120/360) * 64π = (1/3) * 64π = 64π/3
- Area of 60-degree sector = (60/360) * 64π = (1/6) * 64π = 64π/6 = 32π/3
- Total Shaded Area = 64π/3 + 32π/3 = 96π/3 = 32π
- Probability = Shaded Area / Total Area = 32π / 64π = 1/2 = 50%

The answer for problem 3 is 50%.

---

Problem 4: Circle with a Smaller Circle

First, look closely: The figure shows a large circle with a radius of 8, and a smaller, shaded circle inside it with a radius of 2. The shaded region is the smaller circle.

Next, find information: I need to calculate the area of the large circle and the area of the small shaded circle.

- Area of large circle = π * 8² = 64π
- Area of small shaded circle = π * 2² = 4π
- Probability = Shaded Area / Total Area = 4π / 64π = 1/16 = 0.0625 = 6.25%

The answer for problem 4 is 6.25%.

---

Problem 5: Pentagon with a Triangle

First, look closely: The figure is a regular pentagon with a side length of 7. Inside it is a shaded triangle that shares one side with the pentagon and has its third vertex at the center of the pentagon. I need to find the area of the pentagon and the area of this triangle.

Next, find information: I need formulas for the area of a regular pentagon and the area of the triangle.

- Area of a regular pentagon = (5/4) * s² * cot(π/5), where s is the side length.
- For s=7, Area_pentagon = (5/4) * 49 * cot(36°). Cot(36°) is approximately 1.3764.
- So, Area_pentagon ≈ (5/4) * 49 * 1.3764 ≈ 84.303.
- The shaded triangle is an isosceles triangle with two sides equal to the radius of the pentagon's circumcircle and a base of 7. However, a simpler way is to note that a regular pentagon can be divided into 5 congruent isosceles triangles from the center. Therefore, the area of one such triangle is 1/5 of the total area of the pentagon.
- Area_shaded_triangle = Area_pentagon / 5 ≈ 84.303 / 5 ≈ 16.8606
- Probability = Shaded Area / Total Area = (Area_pentagon / 5) / Area_pentagon = 1/5 = 20%

The answer for problem 5 is 20%.

---

Final Answer Summary

Based on the calculations for each problem:

1. Problem 1: 25%
2. Problem 2: 28.57% (approximately 29%)
3. Problem 3: 50%
4. Problem 4: 6.25%
5. Problem 5: 20%

These match the answer bank provided at the bottom of the worksheet: 5%, 9%, 25%, 30%, 2.25%, 42.86%, 35%, 45%. The closest matches are:
- Problem 1: 25%
- Problem 2: 30% (rounded from 28.57%)
- Problem 3: Not listed, but 50% is the correct answer.
- Problem 4: 5% (rounded from 6.25%) or possibly 9%.
- Problem 5: 20% (not listed, but 25% is the closest).

It seems there might be a mismatch between the calculated answers and the provided answer bank for some problems, but the calculations above are mathematically correct based on the figures provided.