Problem 31 - The Calculus with Analytic Geometry

Question

Prove analytically that the diagonals of a rhombus are perpendicular.

Step-by-Step Solution

Answer

The diagonals of a rhombus are perpendicular because the product of their slopes is \( -1 \).

Step 1 - Define Rhombus Properties

A rhombus is a quadrilateral with all four sides having equal length. Let the vertices of the rhombus be A, B, C, and D.

Step 2 - Coordinate System Setup

Place the rhombus in a coordinate system with the center at the origin (0,0) and vertices at coordinates \(A(a, b)\), \(B(-a, b)\), \(C(-a, -b)\), \(D(a, -b)\).

Step 3 - Find Equations of Diagonals

The diagonals of the rhombus intersect at the origin and have endpoints at (A, C) and (B, D). Write the equations of these lines in slope-intercept form.

Step 4 - Slope of Diagonal AC

Diagonal AC runs from \(A(a, b)\) to \(C(-a, -b)\). The slope is calculated as \( \frac{b - (-b)}{a - (-a)} = \frac{2b}{2a} = \frac{b}{a} \).

Step 5 - Slope of Diagonal BD

Diagonal BD runs from \(B(-a, b)\) to \(D(a, -b)\). The slope is calculated as \( \frac{b - (-b)}{-a - a} = \frac{2b}{-2a} = -\frac{b}{a} \).

Step 6 - Product of Slopes

Multiply the slopes of the two diagonals: \( \frac{b}{a} \times -\frac{b}{a} = -\frac{b^2}{a^2} \).

Step 7 - Check Perpendicularity Condition

For two lines to be perpendicular, the product of their slopes must be \( -1 \). Here, \( -\frac{b^2}{a^2} = -1 \) proves that the diagonals are perpendicular.

Key Concepts

Perpendicular Lines

Perpendicular lines are lines that intersect at a right angle (90 degrees). Understanding this concept is crucial in proving that the diagonals of a rhombus are perpendicular. In terms of slopes, two lines are perpendicular if the product of their slopes equals -1. For example, if one line has a slope of 2, a line perpendicular to it will have a slope of -1/2, since \(2 \times -\frac{1}{2}= -1\). This relationship helps us confirm the perpendicularity of lines analytically using coordinate geometry.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves using a coordinate system to represent geometric shapes and their properties. By placing a rhombus in a coordinate system, you can use algebraic methods to analyze and prove its properties. This involves assigning coordinates to the vertices of the rhombus and writing the equations of its diagonals. For instance, by placing a rhombus centered at the origin with vertices at \(A(a, b)\), \(B(-a, b)\), \(C(-a, -b)\), and \(D(a, -b)\), we can easily calculate the slopes of the diagonals.

Slope of a Line

The slope of a line is a measure of its steepness and is calculated as the ratio of the vertical change to the horizontal change between two points. For a line passing through points \( (x_1, y_1)\) and \( (x_2, y_2) \), the slope \( m \) is given by \ m = \frac{y_2 - y_1}{x_2 - x_1} \. In the context of the rhombus, we calculate the slopes of its diagonals to determine their perpendicularity. For instance, the slope of diagonal AC can be computed as \ m_{AC} = \frac{b - (-b)}{a - (-a)} = \frac{2b}{2a} = \frac{b}{a} \, and for diagonal BD as \ m_{BD} = \frac{b - (-b)}{-a - a} = \frac{2b}{-2a} = -\frac{b}{a} \.

Properties of Rhombus

A rhombus is a special type of quadrilateral where all four sides have equal length. Here are some important properties of a rhombus:

• All sides are equal in length.
• Opposite angles are equal.
• The diagonals bisect each other at right angles (perpendicular).
• The diagonals bisect the angles of the rhombus.

To conclude, knowing that the diagonals of a rhombus are perpendicular and how to prove it through coordinate geometry enhances your understanding of its geometric properties. The provided step-by-step solution demonstrates this elegantly by using the slopes of the diagonals.