Problem 27 - Calculus: Early Transcendentals

Question

Find the fraction of the area of a triangle that is occupied by the largest rectangle that can be drawn in the triangle (with one of its sides along a side of the triangle). Show that this fraction does not depend on the dimensions of the given triangle.

Step-by-Step Solution

Answer

The fraction is \(\frac{1}{2}\), and it does not depend on the dimensions of the triangle.

Step 1: Understand the Problem

We need to find the fraction of the area of a triangle that is covered by the largest rectangle which fits inside it, with one side of the rectangle lying along one side of the triangle. We also need to show that this fraction is invariant with respect to the dimensions and shape of the triangle.

Step 2: Analyze a Right Triangle Case

Consider a right triangle first, as this often simplifies initial analysis. Place the triangle on the coordinate plane with the right angle at the origin, the base along the x-axis, and the height along the y-axis. Let the base be length \(b\) and the height be \(h\).

Step 3: Determine Rectangle Dimensions

The largest inscribed rectangle with one side along the base of the triangle will have height equal to half of the triangle's height. So, the rectangle will have a base \(b\) and a height of \(\frac{h}{2}\).

Step 4: Calculate Areas

The area of the triangle is \( \frac{1}{2}bh \). The area of the largest rectangle, which has base \(b\) and height \(\frac{h}{2}\), is \( b \times \frac{h}{2} = \frac{bh}{2} \).

Step 5: Compute the Fraction

The fraction of the triangle's area occupied by the rectangle can be found by dividing the area of the rectangle by the area of the triangle: \[ \text{Fraction} = \frac{\frac{bh}{2}}{\frac{1}{2} bh} = \frac{1}{2}. \]

Step 6: Generalize to Any Triangle

For any triangle, by similar geometric reasoning, the largest rectangle that fits with one side along a side of the triangle and parallel to its opposite side also occupies \(\frac{1}{2}\) of the area of the triangle. This reasoning holds regardless of whether the triangle is scalene, right, or isosceles.

Key Concepts

Triangle Area

The area of a triangle is a fundamental concept in geometry, providing insight into the size and space that the triangle occupies. To calculate the area, you multiply the base of the triangle by its height and then divide by two: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]This formula applies universally to all types of triangles, whether right, scalene, or isosceles.
Understanding this concept allows you to appreciate how the area changes relative to its base and height, which are the primary dimensions.

• Base: The side of the triangle used in the area calculation.
• Height: The perpendicular distance from the base to the opposite vertex.

This straightforward formula transforms into a powerful tool when tackling more complex geometric problems, such as determining the fraction of the triangle's area occupied by another shape.

Inscribed Rectangle

An inscribed rectangle within a triangle maximizes its area when one of its sides aligns precisely along one side of the triangle. Imagine fitting a rectangle perfectly inside a triangle, touching the base while stretching as far vertically as the triangle allows.
This rectangle's height is limited by half the height of the triangle, and its base spans the entire base of the triangle.

• Height of Rectangle: Half of the triangle's height.
• Base of Rectangle: Equal to the base of the triangle.

With these dimensions, the area of the rectangle can be calculated as:\[ \text{Rectangle Area} = \text{base} \times \frac{1}{2} \times \text{height} \] This geometric setup ensures that the rectangle optimally occupies space within the triangle, leading to insightful calculations about fractions of areas.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is the study of geometric figures through a coordinate system. It's particularly useful for visualizing problems like finding areas within triangles.
Placing a triangle on a coordinate plane allows straightforward calculation of distances and areas using algebraic expressions. For a right triangle:

• Right angle position: at the origin (0,0)
• Base: along the x-axis
• Height: along the y-axis

By adopting this approach, you gain the advantage of using coordinates to express geometric relationships algebraically. This simplification aids in calculating the dimensions and areas of inscribed shapes, like rectangles, easing the process of solving geometric problems.

Geometric Reasoning

Geometric reasoning is the process of drawing logical conclusions about geometric figures through properties and relationships. It's the keystone of solving problems such as the fraction of a triangle's area occupied by an inscribed rectangle.
By examining a problem from different angles—literally and figuratively—you can uncover invariant truths regardless of specific measurements.

• Case Analysis: Starting with a specific triangle type like a right triangle simplifies initial calculations.
• Generalization: Extending conclusions to all triangles through similarities and proportional reasoning.

In this exercise, geometric reasoning shows that the largest rectangle fitting in a triangle always occupies half the triangle's area, irrespective of the triangle's size or type. This invariance arises from the consistent way triangles and rectangles interact within these confines, emphasizing the elegance of mathematical patterns.