Question

Show that the rectangle with maximum perimeter that can be inscribed in a circle is a square.

Short answer

The rectangle with maximum perimeter inscribed in a circle is a square, with each side \( r\sqrt{2}/2 \).

Step-by-step solution

Understand the Problem

We need to show that when a rectangle is inscribed in a circle, making its perimeter maximum, it forms a square. For an inscribed rectangle, all its corners will touch the circumference of the circle.

Set Up Equations

Let's consider a circle with radius \( r \). The diagonal of the rectangle inscribed in this circle is equal to the diameter of the circle, which is \( 2r \). Let's denote the sides of the rectangle as \( a \) and \( b \). Thus, we have the relation \( \sqrt{a^2 + b^2} = 2r \).

Express the Perimeter

The perimeter \( P \) of the rectangle is given by \( P = 2a + 2b = 2(a + b) \). Our goal is to find the values of \( a \) and \( b \) that maximize this perimeter while satisfying the constraint from Step 2.

Utilize Constraint for Maximization

From \( \sqrt{a^2 + b^2} = 2r \), square both sides to obtain \( a^2 + b^2 = (2r)^2 = 4r^2 \). Now, to maximize \( a + b \), we'll use the method of Lagrange multipliers or simply use geometry.

Geometry Analysis

Since the diagonal divides the rectangle into two congruent right triangles, a symmetry argument suggests that for maximum perimeter, \( a \) and \( b \) should be equal because the hypotenuse is fixed and maximum at equal sides. Therefore, let \( a = b \).

Solve for \( a \) and \( b \) When Equal

If \( a = b \), substitute into the constraint equation: \( 2a = 2r \), or \( a = r\sqrt{2} \). Therefore, each side is equal, confirming a square, with \( a = b = r\sqrt{2}/2 \).

Derive Maximum Perimeter

For a square, the perimeter becomes \( P = 4a = 4(\frac{r\sqrt{2}}{2}) = 2r\sqrt{2} \), which gives the maximum perimeter for any rectangle with the given conditions.

Key concepts

Rectangle

A rectangle is a four-sided shape with opposite sides that are equal and parallel to each other. It forms a right angle of 90 degrees at each vertex. This characteristic makes rectangles a common figure in geometry.

When dealing with rectangles, a few essential properties are important to remember:

• The opposite sides are of equal length.
• All angles are right angles.
• The diagonals of a rectangle are equal in length.

In the exercise, a rectangle is inscribed inside a circle, meaning all its vertices are on the circle's circumference. This setup presents special conditions where the diagonal of the rectangle must equal the diameter of the circle. As you explore more, you'll see that the rectangle can become a square under the right conditions, especially when maximizing the perimeter.

Circle

A circle is a round plane figure where every point is equidistant from its center. This consistent distance is known as the radius ( r ). These aspects make the circle a fascinating shape with unique properties that often feature in various geometric problems.

Key properties of a circle include:

• Circumference, the circle's boundary, defined by the formula: \( 2\pi r \).
• Diameter, the longest straight line possible within the circle, stretching from one point on the circumference to another, passing through the center. It is twice the radius: \( 2r \).
• An inscribed figure, such as a rectangle, will have its vertices on the circle's circumference.

In the given task, maximizing the rectangle's perimeter involves leveraging the circle's diameter. The diagonal of the rectangle is the same as the circle's diameter, which is a crucial point in the geometry challenge.

Square

A square is a special type of rectangle where all sides are equal in length and every angle is a right angle. Due to its equal sides and angles, a square has unique symmetry properties that come into play in many geometric scenarios.

Here's what stands out about a square:

• All four sides are equal: \( a = b \).
• All angles are 90 degrees.
• Its diagonals are equal and bisect each other at right angles.

In the context of the exercise, for a rectangle with maximum perimeter inscribed in a circle, it must transform into a square. This revelation stems from recognizing that with a fixed diagonal (circle's diameter), the perimeter reaches its highest potential when all sides of the rectangle are equal, effectively becoming a square.

Perimeter

The perimeter of a geometric figure is the total length of its boundary. For a rectangle, perimeter calculation involves summing up all its sides. Specifically, you use the formula:

• Rectangle Perimeter: \( P = 2(a + b) \), where \( a \) and \( b \) are the lengths of the rectangle's sides.
• For squares, the formula simplifies to: \( P = 4a \), since all sides are equal.

When a rectangle is inscribed in a circle, maximizing its perimeter under the condition that its diagonal is the circle's diameter intrduces a fascinating geometric investigation. It's discovered that the rectangle needs to be a square to achieve the largest possible perimeter, highlighting the interplay between the inscribed figure and the circle's properties.