This worksheet covers geometric probability concepts including points on a line, spinner angles, and area ratios within rectangles.

Here are the step-by-step solutions for each problem on the worksheet.

Part 1: Points on a Line Segment

First, find the total length of the segment $\overline{EH}$.
Total Length = $5 + 1 + 2 = 8$.

1. The point is on $\overline{FH}$.
* Find the length of $\overline{FH}$: $FG + GH = 1 + 2 = 3$.
* Probability = $\frac{\text{Length of } FH}{\text{Total Length}} = \frac{3}{8}$.
* Decimal: $0.375$ (rounds to 0.38)

2. The point is not on $\overline{EF}$.
* This means the point is anywhere else on the line ($\overline{FG}$ or $\overline{GH}$).
* Length not on $\overline{EF}$: $1 + 2 = 3$.
* Probability = $\frac{3}{8}$.
* Decimal: $0.375$ (rounds to 0.38)

3. The point is on $\overline{EF}$ or $\overline{GH}$.
* Add the lengths of these two sections: $5 (\text{for } EF) + 2 (\text{for } GH) = 7$.
* Probability = $\frac{7}{8}$.
* Decimal: 0.88 (rounded from 0.875)

4. The point is on $\overline{EG}$.
* Find the length of $\overline{EG}$: $EF + FG = 5 + 1 = 6$.
* Probability = $\frac{6}{8}$, which simplifies to $\frac{3}{4}$.
* Decimal: 0.75

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Part 2: Spinner Probabilities

A full circle is $360^\circ$. To find the probability, divide the angle of the section by 360.

5. The pointer landing on $135^\circ$.
* Fraction: $\frac{135}{360}$
* Simplify: Both divide by 45 $\rightarrow \frac{3}{8}$.
* Decimal: 0.38 (rounded from 0.375)

6. The pointer landing on $75^\circ$.
* Fraction: $\frac{75}{360}$
* Simplify: Both divide by 15 $\rightarrow \frac{5}{24}$.
* Decimal: $5 \div 24 \approx 0.2083...$ rounds to 0.21

7. The pointer landing on $90^\circ$ or $75^\circ$.
* Add the angles: $90 + 75 = 165^\circ$.
* Fraction: $\frac{165}{360}$
* Simplify: Both divide by 15 $\rightarrow \frac{11}{24}$.
* Decimal: $11 \div 24 \approx 0.4583...$ rounds to 0.46

8. The pointer landing on $30^\circ$.
* Fraction: $\frac{30}{360}$
* Simplify: $\frac{3}{36} = \frac{1}{12}$.
* Decimal: $1 \div 12 \approx 0.0833...$ rounds to 0.08

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Part 3: Shapes Inside Rectangles

Formula: $\text{Probability} = \frac{\text{Area of Shape}}{\text{Area of Rectangle}}$

9. The square.
* Area of large rectangle: $7 \text{ in} \times 4 \text{ in} = 28 \text{ sq in}$.
* Area of small square: $2 \text{ in} \times 2 \text{ in} = 4 \text{ sq in}$.
* Probability: $\frac{4}{28} = \frac{1}{7}$.
* Decimal: $1 \div 7 \approx 0.1428...$ rounds to 0.14

10. The triangle.
* Area of large rectangle: $14 \text{ cm} \times 13 \text{ cm} = 182 \text{ sq cm}$.
* Area of triangle: $\frac{1}{2} \times \text{base} \times \text{height} = 0.5 \times 12 \times 5 = 30 \text{ sq cm}$.
* Probability: $\frac{30}{182}$.
* Decimal: $30 \div 182 \approx 0.1648...$ rounds to 0.16

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Part 4: Complex Shapes

First, calculate the total area of the large outer rectangle.
* Total Area = $7 \text{ cm} \times 5 \text{ cm} = 35 \text{ sq cm}$.

11. The triangle.
* Base = 3 cm, Height = 4 cm.
* Area = $0.5 \times 3 \times 4 = 6 \text{ sq cm}$.
* Probability: $\frac{6}{35}$.
* Decimal: $6 \div 35 \approx 0.1714...$ rounds to 0.17

12. The square.
* Side = 2 cm.
* Area = $2 \times 2 = 4 \text{ sq cm}$.
* Probability: $\frac{4}{35}$.
* Decimal: $4 \div 35 \approx 0.1142...$ rounds to 0.11

13. The triangle or the square.
* Add the areas together: $6 (\text{triangle}) + 4 (\text{square}) = 10 \text{ sq cm}$.
* Probability: $\frac{10}{35}$.
* Simplify: $\frac{2}{7}$.
* Decimal: $2 \div 7 \approx 0.2857...$ rounds to 0.29

14. Not the triangle.
* Subtract the triangle's area from the total area: $35 - 6 = 29 \text{ sq cm}$.
* Probability: $\frac{29}{35}$.
* Decimal: $29 \div 35 \approx 0.8285...$ rounds to 0.83

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Final Answer:
1. 0.38
2. 0.38
3. 0.88
4. 0.75
5. 0.38
6. 0.21
7. 0.46
8. 0.08
9. 0.14
10. 0.16
11. 0.17
12. 0.11
13. 0.29
14. 0.83