Geometric probability problems with shaded regions in different shapes.

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

Problem 1: Right Triangle

Goal: Find the probability that a point lies in the shaded region.

1. Find the Total Area:
The total shape is a large right triangle.
* Base = $12$
* Height = $9$
* $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$
* $\text{Total Area} = \frac{1}{2} \times 12 \times 9 = 6 \times 9 = 54$

2. Find the Shaded Area:
The shaded region is a smaller triangle at the top.
* Base = $4$
* Height = $3$
* $\text{Shaded Area} = \frac{1}{2} \times 4 \times 3 = 2 \times 3 = 6$

3. Calculate Probability:
* $P(\text{shaded}) = \frac{\text{Shaded Area}}{\text{Total Area}}$
* $P(\text{shaded}) = \frac{6}{54}$
* Simplify the fraction by dividing top and bottom by 6: $\frac{1}{9}$

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Problem 2: Trapezoid

Goal: Find the probability that a point lies in the shaded regions.

1. Find the Total Area:
The total shape is a trapezoid.
* Top Base ($b_1$) = $20$
* Bottom Base ($b_2$) = $8$
* Height ($h$) = $4$
* $\text{Area} = \frac{1}{2}(b_1 + b_2)h$
* $\text{Total Area} = \frac{1}{2}(20 + 8)4 = \frac{1}{2}(28)4 = 14 \times 4 = 56$

2. Find the Shaded Area:
The shaded regions are two triangles on the sides. Together, their bases make up the difference between the top and bottom widths.
* Combined Base of shaded triangles = $\text{Top Base} - \text{Bottom Base} = 20 - 8 = 12$
* Height = $4$
* $\text{Shaded Area} = \frac{1}{2} \times \text{Combined Base} \times \text{Height}$
* $\text{Shaded Area} = \frac{1}{2} \times 12 \times 4 = 6 \times 4 = 24$

3. Calculate Probability:
* $P(\text{shaded}) = \frac{24}{56}$
* Simplify by dividing top and bottom by 8: $\frac{3}{7}$

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Problem 3: Circle with Sector

Goal: Find the probability that a point lies in the shaded sector.

*Note: In geometric probability for circles, we compare angles or areas. Since the radius is the same for both, we can just compare the angles.*

1. Identify Angles:
* Angle of shaded region = $118^\circ$
* Total angle of a circle = $360^\circ$

2. Calculate Probability:
* $P(\text{shaded}) = \frac{118}{360}$
* Simplify by dividing top and bottom by 2: $\frac{59}{180}$

*(Note: If calculating by area using radius $r=8$: Total Area $= 64\pi$, Shaded Area $= \frac{118}{360} \times 64\pi$. The $\pi$ and $64$ cancel out, leaving the same ratio $\frac{118}{360}$.)*

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Problem 4: Concentric Circles

Goal: Find the probability that a point lies in the shaded ring (annulus).

1. Find Dimensions:
* Inner Radius ($r$) = $3$
* Outer Radius ($R$) = $5$ (indicated by the arrow spanning from center to edge)

2. Find Areas:
* $\text{Total Area} = \pi R^2 = \pi(5)^2 = 25\pi$
* $\text{Inner White Area} = \pi r^2 = \pi(3)^2 = 9\pi$
* $\text{Shaded Area} = \text{Total Area} - \text{Inner White Area}$
* $\text{Shaded Area} = 25\pi - 9\pi = 16\pi$

3. Calculate Probability:
* $P(\text{shaded}) = \frac{16\pi}{25\pi}$
* Cancel $\pi$: $\frac{16}{25}$

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Problem 5: Pentagon

Goal: Find the probability that a point lies in the shaded triangle.

1. Analyze the Shape:
The figure shows a regular pentagon divided into 5 congruent (identical) triangles meeting at the center.

2. Calculate Probability:
* There are 5 equal parts in total.
* 1 part is shaded.
* $P(\text{shaded}) = \frac{1}{5}$

Final Answer:
1. 1/9
2. 3/7
3. 59/180
4. 16/25
5. 1/5