https://file.aiquickdraw.com/mathbot/file/1729935727652wktnugqg.jpg

The picture shows a rectangle inside of a 90 - degree triangle. What is the biggest areal the rectangle can have?

To determine the largest possible area of a rectangle inscribed in a right triangle, let's follow these steps:

### Step 1: Identify Triangle Dimensions
The triangle is a right triangle with legs of lengths:
- **30** (one leg)
- **40** (the other leg)

Using these two sides, we can calculate the hypotenuse \( c \) of the triangle using the Pythagorean theorem:
\[
c = \sqrt{30^2 + 40^2} = \sqrt{900 + 1600} = \sqrt{2500} = 50
\]

### Step 2: Largest Inscribed Rectangle in a Right Triangle
In a right triangle, the largest area for an inscribed rectangle occurs when the rectangle’s height and width are proportional to the legs of the triangle. The maximum area \( A \) for an inscribed rectangle can be calculated as:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
where "base" and "height" are the legs of the triangle.

### Step 3: Calculate Maximum Rectangle Area
For this triangle, using the legs \( 30 \) and \( 40 \), the maximum area of the rectangle is:
\[
A = \frac{1}{2} \times 30 \times 40 = \frac{1}{2} \times 1200 = 600
\]

### Conclusion
The largest possible area of the rectangle that can be inscribed in this right triangle is:
\[
\boxed{600}
\]

Would you like more detailed steps or have questions? 

### Related Questions
1. How would the area change if the triangle's legs were different lengths?
2. What is the formula for the hypotenuse in any right triangle?
3. How does the area of a rectangle inscribed in an isosceles right triangle compare to this one?
4. Can a rectangle inscribed in any triangle achieve half the area of the triangle?
5. How do we derive the formula for the largest inscribed rectangle in any triangle?

### Tip
In problems involving inscribed rectangles in triangles, the maximum area occurs when the rectangle’s dimensions maintain the same proportions as the triangle's legs.

The picture shows a rectangle inside of a 90-degree triangle. What is the biggest area the rectangle can have?

Explore how to find the largest possible area of a rectangle inscribed within a right triangle with legs 30 and 40. This solution includes the use of the Pythagorean theorem and optimization techniques, suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation