The Largest Area of a Triangle Inside a Rectangle

When considering the placement of the largest possible triangle within a given rectangle with length ( l ) and width ( w ), it is a relevant task in both geometry and applicable to optimization problems in various fields. This article explores the mathematical basis for determining the maximum area of such a triangle and discusses the conditions under which this maximum area is achieved.

Introduction

Given a rectangle with dimensions ( l ) and ( w ), the largest triangle that can be inscribed in this rectangle follows a specific geometric configuration. This configuration involves using the geometry of the rectangle and the formula for the area of a triangle to find the optimal area of the inscribed triangle.

Maximum Area of a Triangle in a Rectangle

The largest possible area of a triangle that can be inscribed in a rectangle with length ( l ) and width ( w ) is:

[text{Area} frac{l times w}{2}]

This result can be derived from understanding the geometric properties of a rectangle and the inscribed triangle. The key insight is that the largest triangle is formed by one of the rectangle's sides as the base and the opposite side as the height.

Proof and Explanation

To prove this, let's consider the possible configurations of the triangle within the rectangle:

No vertex of the triangle coincides with a vertex of the rectangle and all three vertices lie on different sides:

In this case, the triangle is smaller, as it does not utilize the full length or width of the rectangle. Therefore, the area of such a triangle will be less than (frac{l times w}{2}).

Exactly one vertex of the triangle coincides with a vertex of the rectangle:

This is the standard case mentioned in the original problem. If one vertex of the triangle lies on a vertex of the rectangle, the triangle can be configured such that one of its sides is the length or width of the rectangle. The maximum area of this triangle is (frac{l times w}{2}).

More than one vertex of the triangle coincide with vertices of the rectangle:

This configuration yields an area of (frac{l times w}{2}) as well. If two vertices of the triangle lie on the vertices of the rectangle, the third vertex must be placed on the opposite side, maximizing the area.

For example, if we take the triangle ( triangle EKC ) within the rectangle ( ABCD ) where ( E ) and ( K ) are vertices of the rectangle, the area of this triangle is maximized when the base is either ( l ) or ( w ) and the height is the other dimension. Therefore, the area is calculated as:

[text{Area} frac{1}{2} times l times w]

Conclusion

Solving the problem of finding the largest area of a triangle within a rectangle involves understanding the geometric properties and the application of the formula for the area of a triangle. The maximum area is achieved in configurations where one side of the triangle is one of the sides of the rectangle, and the opposite vertex is on the opposite side. By using these configurations, we can deduce that the largest area of a triangle that can be inscribed in a rectangle with length ( l ) and width ( w ) is (frac{l times w}{2}).