Problem 54 - Precalculus

Question

This exercise outlines a proof of the fact that two nonvertical lines with slopes \(m_{1}\) and \(m_{2}\) are perpendicular if and only if \(m_{1} m_{2}=-1 .\) In the following figure, we've assumed that our two nonvertical lines \(y=m_{1} x\) and \(y=m_{2} x\) intersect at the origin. [If they did not intersect there, we could just as well work with lines parallel to these that do intersect at \((0,0),\) recalling that parallel lines have the same slope.] The proof relies on the following geometric fact: \(\overline{O A} \perp \overline{O B} \quad\) if and only if \(\quad(O A)^{2}+(O B)^{2}=(A B)^{2}\). (a) Verify that the coordinates of \(A\) and \(B\) are \(A\left(1, m_{1}\right)\) and \(B\left(1, m_{2}\right)\) (b) Show that $$\begin{aligned}O A^{2} &=1+m_{1}^{2} \\\O B^{2} &=1+m_{2}^{2} \\\A B^{2} &=m_{1}^{2}-2 m_{1} m_{2}+m_{2}^{2} \end{aligned}$$ (c) Use part (b) to show that the equation $$O A^{2}+O B^{2}=A B^{2}$$ is equivalent to \(m_{1} m_{2}=-1\) (GRAPH CAN'T COPY)

Step-by-Step Solution

Answer

The slopes \( m_1 \) and \( m_2 \) are perpendicular if \( m_1m_2 = -1 \).

Step 1: Find Coordinates of Points A and B

Since the lines intersect at the origin, the coordinates of point \( A \) on line \( y = m_1 x \) at \( x = 1 \) are \( A(1, m_1) \). Similarly, the coordinates of point \( B \) on line \( y = m_2 x \) at \( x = 1 \) are \( B(1, m_2) \).

Step 2: Calculate the Squares of OA and OB

Using the Pythagorean Theorem, compute the lengths of \( OA \) and \( OB \). Since \( O = (0, 0) \): 1. \( OA = \sqrt{(1-0)^2 + (m_1-0)^2} = \sqrt{1 + m_1^2} \), thus \( OA^2 = 1 + m_1^2 \).2. \( OB = \sqrt{(1-0)^2 + (m_2-0)^2} = \sqrt{1 + m_2^2} \), thus \( OB^2 = 1 + m_2^2 \).

Step 3: Calculate the Square of AB

The distance \( AB \) is calculated using the formula for distance between two points: \[ AB = \sqrt{(1-1)^2 + (m_2-m_1)^2} = \sqrt{(m_2 - m_1)^2} \]. Thus:\( AB^2 = (m_2 - m_1)^2 = m_2^2 - 2m_1m_2 + m_1^2 \).

Step 4: Establish the Equation OA² + OB² = AB²

Using the results from Steps 2 and 3, plug into the equation:\( OA^2 + OB^2 = A B^2 \) becomes \( (1 + m_1^2) + (1 + m_2^2) = m_1^2 - 2m_1m_2 + m_2^2 \).

Step 5: Simplify and Solve the Equation

Simplify the equation from Step 4:\( 2 + m_1^2 + m_2^2 = m_1^2 - 2m_1m_2 + m_2^2 \)Cancelling \( m_1^2 \) and \( m_2^2 \) from both sides leaves \( 2 = -2m_1m_2 \). Solving for \( m_1m_2 \) gives \( m_1m_2 = -1 \).

Key Concepts

Slope of a Line

The slope of a line is a measure of its steepness and direction. It is represented by the letter \( m \) and can be calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \((x_1, y_1)\) and \((x_2, y_2)\) are two points on the line.
The concept of slope helps in understanding the relationship between different lines. Two lines are said to be perpendicular if the product of their slopes is \(-1\). This is a key idea for proving the relationship between perpendicular lines. In our exercise, two nonvertical lines intersecting at the origin have slopes \( m_1 \) and \( m_2 \), and proving that they are perpendicular involves showing that \( m_1m_2 = -1 \).
Understanding the slope is crucial for many geometric applications, including identifying parallel and perpendicular lines, analyzing graphs, and solving optimization problems.

Geometric Proof

A geometric proof involves using established geometric principles to demonstrate a given proposition or theorem. In our exercise, the aim is to prove that two lines with certain slopes are perpendicular. The geometric proof relies on verifying a specific property involving distances calculated using the Pythagorean theorem.
Our exercise demonstrates this by using the relationship \( \overline{OA} \perp \overline{OB} \) if and only if \( (OA)^2 + (OB)^2 = (AB)^2 \). This relation helps to connect the slopes of the lines \( m_1 \) and \( m_2 \) to their perpendicularity. The aim is to demonstrate this by calculating the specific distances and verifying that the condition is satisfied.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Mathematically, it is expressed as \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse. In coordinate geometry, this theorem is often used to find the distance between two points.
In our exercise, the Pythagorean theorem is used to calculate the lengths \( OA \), \( OB \), and \( AB \), which are the distances from the origin to points \( A \) and \( B \), and between points \( A \) and \( B \), respectively. Using this theorem allows us to establish the necessary equations to prove the lines' perpendicularity.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. It allows us to solve geometric problems algebraically by assigning coordinates to geometric shapes and using equations to describe their relationships.
In the context of our exercise, coordinate geometry helps in understanding the placement and properties of lines in a two-dimensional space. By specifying the coordinates of points \( A(1, m_1) \) and \( B(1, m_2) \), we can use equations to deduce properties like slopes, distances, and relationships such as perpendicularity.
Coordinate geometry is a powerful tool in modern mathematics, enabling precise measurements and insights into geometric figures that can be cumbersome to explore with traditional methods alone.