Problem 95 - Contemporary Precalculus

Question

Let \((c, d)\) be any point in the plane with \(c \neq 0 .\) Prove that \((c, d)\) and \((-c,-d)\) lie on the same straight line through the origin, on opposite sides of the origin, the same distance from the origin. [Hint: Find the midpoint of the line segment joining \((c, d) \text { and }(-c,-d) .]\)

Step-by-Step Solution

Answer

Question: Given two points (c, d) and (-c, -d) in the plane with c ≠ 0, prove that these two points lie on the same straight line through the origin, on opposite sides of the origin, and the same distance from the origin. Answer: The statement has been proven through the following steps: First, we found the slope of the line passing through the given points to be m = d/c. Next, we derived the equation of the line using the point-slope form, resulting in y = d(x/c - 1). After verifying that the origin (0,0) lies on this line, we determined the midpoint between the given points as M(0,0), indicating that the points are equidistant from the origin. Thus, the points (c, d) and (-c, -d) lie on the same straight line through the origin, on opposite sides of the origin, and at the same distance from the origin.

Step 1: Find the slope of the line

To find the slope of the line passing through two points \((x_1, y_1) = (c, d)\) and \((x_2, y_2) = (-c, -d)\), we can use the formula: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$. So, in our case, the slope will be: $$m = \frac{-d - d}{-c- c} = \frac{-2d}{-2c} = \frac{d}{c}$$.

Step 2: Find the equation of the line

Using the point-slope form of the equation of a line: \(y - y_1 = m (x - x_1)\), and substituting the point \((c,d)\) and slope \(m = d/c\), we have: \(y-d = \frac{d}{c} (x-c)\). Simplifying it, the equation of the line is: $$y= d\left(\frac{x}{c}-1\right)$$.

Step 3: Verify that \((0,0)\) (the origin) is on the line

To check if the origin lies on the line, substitute \(x = 0\) and \(y = 0\) into the line equation: $$0= d\left(\frac{0}{c}-1\right)$$. Since \(c\) is nonzero, the equality holds and the origin is on the line.

Step 4: Find the midpoint of the line segment

To find the midpoint between the points \((c, d)\) and \((-c,-d)\), we need to find the average of the x-coordinates and the y-coordinates. The midpoint is given by: $$M\left(\frac{c + (-c)}{2}, \frac{d + (-d)}{2}\right) = M(0,0)$$. So the midpoint is the origin (0,0).

Step 5: Conclude the proof

Based on our findings, we have shown that points \((c, d)\) and \((-c, -d)\) both lie on the line passing through the origin and are at an equal distance from the origin (measured by the midpoint). Therefore, we have proven the statement that \((c, d)\) and \((-c, -d)\) lie on the same straight line through the origin, on opposite sides of the origin, and the same distance from the origin.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines algebra and geometry to create a system where geometric shapes can be described and analyzed through algebraic equations. This field is fundamental to many aspects of mathematics, as it allows for a concrete representation of abstract geometric concepts.

In the problem provided, coordinate geometry is used to prove the relationship between points and their position relative to the origin in the Cartesian plane. By plotting points \( (c, d) \) and \( (-c, -d) \) and finding the midpoint, we apply the principles of coordinate geometry to demonstrate that these points lie on a straight line passing through the origin. This line is defined by a linear equation derived from the coordinates of the points and their slope.

Slope of a Line

The slope of a line is a measure of its steepness and direction. It's a core concept in coordinate geometry and is typically denoted as \( m \). The slope can be calculated when given two points \( (x_1, y_1) \) and \( (x_2, y_2) \) on the line, using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

For the given exercise, the slope of the line through points \( (c, d) \) and \( (-c, -d) \) was calculated as \( m = \frac{d}{c} \), signifying a precise rate of change in the 'y' coordinate with respect to the 'x' coordinate. Understanding how to calculate the slope is vital, as it not only describes the line's angle relative to the horizontal but also allows us to write the equation of the line using different forms, such as point-slope form.

Point-Slope Form of a Line

The point-slope form is an equation of a straight line that uses the slope of the line and the coordinates of a particular point it passes through. Given a slope \( m \) and a point \( (x_1, y_1) \) on the line, the point-slope form is written as \( y - y_1 = m (x - x_1) \).

This form is useful when we already know a point on the line and the slope. For instance, with the point \( (c,d) \) and the slope \( d/c \) derived earlier, the point-slope form for the line in question is simplified to \( y = d\left(\frac{x}{c}-1\right) \). Point-slope form makes it straightforward to verify if other points, like the origin, lie on the line or to construct the full equation of the line by simplifying to the 'y = mx + b' format.