Problem 20 - Thomas Calculus

Question

Perpendicular diagonals Show that squares are the only rectangles with perpendicular diagonals.

Step-by-Step Solution

Answer

Squares are the only rectangles with perpendicular diagonals.

Step 1: Define a Rectangle's Properties

A rectangle is a quadrilateral with opposite sides equal and each interior angle as 90°. Let the lengths of the rectangle be \(a\) and \(b\).

Step 2: Recall the Diagonals of a Rectangle

A rectangle has two diagonals, each of which has the same length. Recall that the length of the diagonal \(d\) is given by the Pythagorean theorem: \[d = \sqrt{a^2 + b^2}\].

Step 3: Condition for Perpendicular Diagonals

For diagonals to be perpendicular, they must satisfy the condition \(d_1 \cdot d_2 = 0\), where \(d_1\) and \(d_2\) are the vectors corresponding to the diagonals.

Step 4: Apply Perpendicular Condition to Rectangle

Using vector notation, the diagonals of a rectangle can be defined as \( (a, b) \) and \( (a, -b) \). The dot product of these diagonals is: \[ (a, b) \cdot (a, -b) = a^2 - b^2 \]. For this to equal zero, we have \(a^2 - b^2 = 0\).

Step 5: Solve for \(a\) and \(b\)

The equation \(a^2 - b^2 = 0\) simplifies to \(a^2 = b^2\), which implies \(a = b\). Thus, the rectangle must have equal sides.

Step 6: Conclude the Shape

If \(a = b\), then the rectangle is equilateral, which is, by definition, a square. Therefore, squares are the only rectangles with perpendicular diagonals.

Key Concepts

Rectangles

A rectangle is a four-sided shape known as a quadrilateral. All rectangles have certain key attributes: opposite sides that are equal in length and all interior angles that each measure 90 degrees. This structure is fundamental to understanding why not all rectangles have perpendicular diagonals. When discussing rectangles, it's essential to clarify a few properties:

• Opposite sides are equal, i.e., if the sides of the rectangle are labeled as 'a' and 'b', then each pair of opposite sides is the same (both a's equal and both b's equal).
• The angles inside a rectangle are right angles, making them perfect candidates for using the Pythagorean theorem.
• Rectangles contain two diagonals, which are line segments joining opposite corners.

These properties contribute to why only squares have perpendicular diagonals, as explored further in relationship with their diagonals and angles.

Squares

One might wonder why squares are unique amongst rectangles with the property of having perpendicular diagonals. Well, a square is a special type of rectangle where not only the opposite sides are equal, but all four sides are equal. This characteristic leads to some interesting mathematical properties:

• Each angle within a square is a right angle, just like in all rectangles.
• All sides being equal implies that the diagonals intersect each other at 90 degrees.
• Since the diagonals are of equal length and meet perpendicularly, they essentially divide the square into four right triangles.

These attributes make the square quite distinct. To satisfy perpendicular diagonal properties, a rectangle must have equal sides, transforming it into a square. This trait of equal sides effortlessly fulfills the perpendicular condition by ensuring the sides' length causes diagonals to bisect angles at right angles, confirming they must be perpendicular.

Pythagorean theorem

The Pythagorean theorem plays a pivotal role in proving why only squares can have perpendicular diagonals. This theorem establishes a relationship between the three sides of a right triangle, stating that the square of the hypotenuse (the longest side) equals the sum of the squares of the other two sides. Mathematically, it’s expressed as:\[c^2 = a^2 + b^2\]In a rectangle, the diagonals act as the hypotenuse when you consider them as forming two right triangles within the shape. This is how we calculate the length of the diagonal: \[d = \sqrt{a^2 + b^2}\] Where 'a' and 'b' are the rectangle's sides.For the diagonals to be perpendicular, certain vector-based calculations lead you back to this - the need for \[a^2 - b^2 = 0\]which ultimately defines equal side lengths meaning the shape must be a square. The theorem provides a foundational formula to explore and affirm proofs of geometric properties, such as this exclusivity of squares with perpendicular diagonals in rectangles.