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What is the biggest area the rectangle can have?

To find the largest area for the rectangle inside the triangle, we can follow these steps:

1. **Identify the triangle's dimensions**: The triangle is a right triangle with legs 30 and 40, which means its hypotenuse can be found using the Pythagorean theorem.

   \[
   \text{Hypotenuse} = \sqrt{30^2 + 40^2} = \sqrt{900 + 1600} = \sqrt{2500} = 50
   \]

2. **Area of the triangle**: The area of a right triangle is given by:

   \[
   \text{Area}_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 30 \times 40 = 600
   \]

3. **Maximum rectangle area inside a right triangle**: The largest possible rectangle that can fit within a right triangle occupies exactly half of the triangle's area. 

   \[
   \text{Max Area}_{\text{rectangle}} = \frac{1}{2} \times \text{Area}_{\text{triangle}} = \frac{1}{2} \times 600 = 300
   \]

Therefore, the largest area the rectangle can have is **300 square units**.

Would you like further details on this approach, or do you have any questions?

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Here are 5 additional questions related to this concept:

1. How would the rectangle's area change if the triangle's dimensions were doubled?
2. Can you derive the formula for the maximum rectangle area inside any right triangle?
3. How does this approach apply if the triangle is not a right triangle?
4. If you were given only the hypotenuse and one leg, how would you calculate the rectangle's maximum area?
5. What is the relationship between the aspect ratio of the triangle and the rectangle's dimensions?

**Tip**: In a right triangle, the largest inscribed rectangle's area is always half of the triangle's area.

Explore how to find the largest possible area of a rectangle that can be inscribed within a right triangle with legs of 30 and 40 units. Using the Pythagorean theorem and area calculation, this problem shows that the maximum rectangle area is half of the triangle’s area, providing a step-by-step solution for high school students.

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