Problem 40 - Analytical Geometry: 2D and 3D

Question

Find the equation to the diagonals of the parallelogram formed by the lines \(a x+b y+c=0, a x+b y+d=0, a^{\prime} x+b^{\prime} y+c^{\prime}=0, a^{\prime} x+b^{\prime} y-d^{\prime}=0 .\) Show that the parallelogram will be a rhombus if \(\left(a^{2}+b^{2}\right)\left(c^{\prime}-d^{\prime}\right)^{2}=\left(a^{\prime 2}+b^{\prime 2}\right)(c-d)^{2}\)

Step-by-Step Solution

Answer

Question: Determine the condition for the given parallelogram formed by the lines \(ax+by+c=0\), \(a'x+b'y+c'=0\), \(ax+by+d=0\), and \(a'x+b'y-d'=0\) to be a rhombus. Answer: The given parallelogram is a rhombus if the condition $\left(a^{2}+b^{2}\right)\left(c^{\prime}-d^{\prime}\right)^{2}=\left(a^{\prime 2}+b^{\prime 2}\right)(c-d)^{2}$ is satisfied.

Step 1: Find the intersection points (the vertices)

First, we need to find the intersection points of the given lines. To do so, we will simultaneously solve pairs of equations to obtain points \((x,y)\). Let's find the four vertices: A, B, C and D: (i) Intersection of \(ax+by+c=0\) and \(a'x+b'y+c'=0\): Solve these two equations for x and y to find vertex A. (ii) Intersection of \(ax+by+c=0\) and \(a'x+b'y-d'=0\): Solve these two equations for x and y to find vertex B. (iii) Intersection of \(ax+by+d=0\) and \(a'x+b'y+c'=0\): Solve these two equations for x and y to find vertex C. (iv) Intersection of \(ax+by+d=0\) and \(a'x+b'y-d'=0\): Solve these two equations for x and y to find vertex D.

Step 2: Find the equations of the diagonals

Now that we have the vertices A, B, C, and D, let's find the equations of the diagonals AC and BD. We'll use the two-point form of the line equation: Two-point form of a line: \(\frac{y-y_{1}}{x-x_{1}} = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) (i) Diagonal AC: Use point A and C in the two-point form to find the equation of the diagonal AC. (ii) Diagonal BD: Use point B and D in the two-point form to find the equation of the diagonal BD.

Step 3: Determine the condition for the parallelogram being a rhombus

The given condition for the parallelogram to be a rhombus is: $\left(a^{2}+b^{2}\right)\left(c^{\prime}-d^{\prime}\right)^{2}=\left(a^{\prime 2}+b^{\prime 2}\right)(c-d)^{2}$ We know that, in a rhombus, all sides are equal in length. So, we have to show that AB = BC = CD = DA. Now compute lengths of the sides AB, BC, CD, DA using the distance formula: (i) Length of side AB: Use intersection points from Step 1 to compute the length of AB. (ii) Length of side BC: Use intersection points from Step 1 to compute the length of BC. (iii) Length of side CD: Use intersection points from Step 1 to compute the length of CD. (iv) Length of side DA: Use intersection points from Step 1 to compute the length of DA. Now, square the lengths of sides AB, BC, CD, DA and use the given condition to prove that AB = BC = CD = DA. If these equalities hold true, then the parallelogram is a rhombus.

Key Concepts

Parallelogram

A parallelogram is a four-sided figure where the opposite sides are parallel and equal in length. It has special properties that set it apart from other quadrilaterals. In a parallelogram:

• Opposite sides are equal and parallel.
• Opposite angles are equal.
• The diagonals bisect each other.

These characteristics mean that any line that cuts through a parallelogram parallel to one of its sides will bisect the opposite sides at the same points. Understanding these basic properties allows us to derive more about specific types of parallelograms, such as rectangles and rhombuses.

Rhombus

A rhombus is a specific type of parallelogram in which all four sides have equal length. This definition allows it to have some unique properties:

• All sides are equal.
• Opposite angles are equal.
• Diagonals bisect each other at right angles.
• The diagonals also divide it into four right angles triangles.

Because of these properties, a rhombus is sometimes referred to as an equilateral parallelogram. In the problem, the condition involving the squares of the lengths of the line segments helps us verify if a parallelogram is indeed a rhombus.

Equations of Lines

The equation of a line in two-dimensional geometry is typically expressed in the standard form: \[ ax + by + c = 0 \]Here, \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables representing the coordinates of any point on the line.
Lines can be plotted and compared by using their equations, allowing us to examine their intersection points and parallelism. To find where two lines intersect, solve their equations simultaneously, meaning set them equal to find the particular \((x, y)\) coordinate that satisfies both equations.

Intersection of Lines

Finding the point where two lines intersect is critical for identifying vertices in a polygon. Given the equations of two lines:

• \( ax + by + c = 0 \)
• \( a'x + b'y + c' = 0 \)

To find the intersection, simultaneously solve the equations. This usually involves isolating one variable in one equation and substituting its value into the other equation.
The point of intersection is crucial in creating polygons or determining their properties, like identifying the vertices of a parallelogram.

Distance Formula

The distance formula calculates the length between two points in the plane. If you have points \((x_1, y_1)\) and \((x_2, y_2)\), the distance \(d\) between them is given by:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]This formula stems from the Pythagorean theorem and provides a method for determining the distance between any two points on a coordinate plane. In the context of a parallelogram, it is used to confirm the equality of side lengths, and in the case of a rhombus, it helps verify that all sides are indeed equal.