To solve the problem, we need to calculate the area of Region A as a fraction of the area of the large triangle, and use that fraction as the probability. Here's the step-by-step breakdown:

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Step 1: Area of the Large Triangle

The vertices of the triangle are $(0, 0)$, $(12, 0)$, and $(0, 8)$. The area of a triangle is given by: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ Here, the base is 12 (distance along the x-axis), and the height is 8 (distance along the y-axis). Thus: $\text{Area of large triangle} = \frac{1}{2} \times 12 \times 8 = 48$

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Step 2: Description of Region A

Region A is a trapezoid, bounded by:

1. The y-axis ($x = 0$),
2. The vertical line $x = 3$,
3. The hypotenuse of the large triangle (sloping line from $(12, 0)$ to $(0, 8)$),
4. The horizontal line $y = 6$.

Trapezoid Dimensions

The key to finding the area of Region A is calculating the lengths of the two parallel sides (heights of the trapezoid):

• The vertical line $x = 0$ extends from $y = 0$ to $y = 8$, so one height is 8.
• The vertical line $x = 3$ extends from $y = 0$ to $y = 6$, so the other height is 6.

The width of the trapezoid (the distance between $x = 0$ and $x = 3$) is 3.

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Step 3: Area of Region A

The area of a trapezoid is given by: $\text{Area} = \frac{1}{2} \times (\text{base}_1 + \text{base}_2) \times \text{height}$ Here, $\text{base}_1 = 8$, $\text{base}_2 = 6$, and $\text{height} = 3$. Thus: $\text{Area of Region A} = \frac{1}{2} \times (8 + 6) \times 3 = \frac{1}{2} \times 14 \times 3 = 21$

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Step 4: Probability

The probability is the ratio of the area of Region A to the area of the large triangle: $\text{Probability} = \frac{\text{Area of Region A}}{\text{Area of Large Triangle}} = \frac{21}{48}$ Simplify: $\text{Probability} = \frac{7}{16} \approx 0.4375 \text{ or } 43.75\%$

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Final Answer

The probability is approximately 44%.

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Follow-up Questions

1. How do we calculate the area of Region B to confirm the result?
2. What would the probability be if Region A was divided further into smaller sections?
3. How does the slope of the hypotenuse affect the division of regions?
4. Can we confirm the dimensions of Region A using algebraic methods?
5. How does the probability change if the dividing line shifts horizontally?

Tip: Always verify your trapezoid dimensions carefully, as small errors can significantly impact area calculations.