Problem 78 - Algebra and Trigonometry

Question

Collinear Points Suppose you are given the coordinates of three points in the plane and you want to see whether they lie on the same line. How can you do this using slopes? Using the Distance Formula? Can you think of another method?

Step-by-Step Solution

Answer

Check if slopes are equal, if distance sums match, or if triangle area is zero.

Step 1: Calculate Slopes

To check collinearity using slopes, calculate the slope between each pair of points. Let's denote the three points as \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \). Calculate the slope \( m_{AB} \) between A and B using the formula \( m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} \). Similarly, calculate the slope \( m_{BC} \) between B and C and \( m_{AC} \) between A and C. If all slopes are equal, the points are collinear.

Step 2: Check Using Distance Formula

The points are collinear if the sum of the distances between two pairs equals the distance of the longest side. Use the distance formula to calculate the length of each segment: \( AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \), \( BC = \sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2} \), and \( AC = \sqrt{(x_3 - x_1)^2 + (y_3 - y_1)^2} \). If \( AB + BC = AC \) or other combinations, then they are collinear.

Step 3: Use the Area of a Triangle Method

Another method involves calculating the area of the triangle formed by the three points. If the area is zero, then the points are collinear. The area can be calculated using the formula \( \text{Area} = \frac{1}{2} | x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) | \). If the result is zero, the points are collinear.

Key Concepts

Calculating Slopes

When we talk about determining if points are collinear, a great starting point is to calculate the slopes between each pair of points. This means determining if they all lie on the same straight line. Begin with three points, say \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \). For two points, the slope is essentially how much the line goes up or down, divided by how much it goes left or right.
Here's how you calculate it:

• Slope \( m_{AB} \): Use the formula \( m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} \).
• Slope \( m_{BC} \): Use the formula \( m_{BC} = \frac{y_3 - y_2}{x_3 - x_2} \).
• Slope \( m_{AC} \): Use the formula \( m_{AC} = \frac{y_3 - y_1}{x_3 - x_1} \).

Keep in mind, if all the slopes, \( m_{AB} \), \( m_{BC} \), and \( m_{AC} \), are equal, the points are collinear.
This means they perfectly align with each other on the same straight path. Important to note, parallel lines have the same slope, which helps us identify such aligned points.

Distance Formula

Another insightful method to check for collinear points involves the Distance Formula, which finds the physical distance between two points. Think of each line segment connecting the points as a separate measurement.
First, calculate the lengths of segments \( AB \), \( BC \), and \( AC \) using the Distance Formula:

• Distance \( AB \): Given as \( AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).
• Distance \( BC \): Calculated by \( BC = \sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2} \).
• Distance \( AC \): Use \( AC = \sqrt{(x_3 - x_1)^2 + (y_3 - y_1)^2} \).

Next, if the sum of the lengths of the two shorter segments equals the length of the longest segment, then the points are collinear.
This is because they all fall on the direct path between the start and endpoint, without veering off into another direction. It's a clear way to measure alignment by physical distances.

Triangle Area Method

Using the Triangle Area Method provides another fascinating approach to establishing the collinearity of points. This method leverages the notion of the area formed by the three given points.
If the computed area of the triangle made by the points is zero, then congrats, the points are collinear.
To calculate the area, use this formula: \[ \text{Area} = \frac{1}{2} | x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) | \]
Put simply:

• You input the x and y values for each point as seen in the formula.
• The vertical bars mean you take the absolute value, ensuring a non-negative area.

When you calculate it fully and the area is zero, it indicates there's no actual triangle because the points are merely stretched in a straight line.
Thus, making it a nifty mathematical check for alignment! This method creatively uses geometry to solve a basic algebraic query.