Problem 74 - Algebra and Trigonometry

Question

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

To check collinearity, compare slopes, calculate distances, or use the area formula.

Step 1: Understanding the Problem

We need to determine if three points, say \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\), lie on the same straight line in a 2D plane. The exercise involves using different methods: slopes, the Distance Formula, and potentially another method to verify collinearity.

Step 2: Using Slopes

Calculate the slope between each pair of points. If the slope between \((x_1, y_1)\) and \((x_2, y_2)\) is \(m_1\) and the slope between \((x_2, y_2)\) and \((x_3, y_3)\) is \(m_2\), then:\[m_1 = \frac{y_2 - y_1}{x_2 - x_1}\]\[m_2 = \frac{y_3 - y_2}{x_3 - x_2}\]If \(m_1 = m_2\), then the points are collinear (i.e., they lie on the same line).

Step 3: Using the Distance Formula

Calculate the distances between each pair of points using the formula:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Let the distances be \(d_{12}\), \(d_{23}\), and \(d_{13}\). If the sum of the two smaller distances equals the largest distance \(d_{12} + d_{23} = d_{13}\), then the points are collinear. Ensure the calculation considers which distance is the longest.

Step 4: Using the Area of the Triangle (Alternative Method)

Use the determinant to find the area of the triangle formed by three points. Compute the area \(A\) using:\[ A = \frac{1}{2} | x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) | \]If \(A = 0\), the points are collinear, as a zero area indicates the points lie on a straight line.

Key Concepts

Slope of a Line

The slope of a line gives us a measure of its steepness and direction. When checking if three points are collinear, we calculate the slope between each pair of points. If these slopes are equal, it shows that all three points lie on the same straight line.

To find the slope between two points, say \((x_1, y_1)\) and \((x_2, y_2)\), we use the formula:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

This formula gives the change in \(y\) divided by the change in \(x\), representing the line's slope. To test collinearity with three points \((x_1, y_1)\), \((x_2, y_2)\), \((x_3, y_3)\):

• Compute the slope between \((x_1, y_1)\) and \((x_2, y_2)\).
• Compute the slope between \((x_2, y_2)\) and \((x_3, y_3)\).
• If these slopes are equal, the points are collinear.

Keep in mind, dividing by zero becomes an issue if the line is vertical, as the x-coordinates are the same, resulting in undefined slope. This consistency across the pairs' slopes confirms alignment on a single line.

Distance Formula

The distance formula helps us calculate how far apart two points are in the plane. This is crucial for determining if three points are collinear using their relative distances. The formula uses the differences in x and y coordinates to find the distance between two points:

• \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

When checking for collinearity using distances:

• Determine the distance between each pair of points: \(d_{12}\), \(d_{23}\), and \(d_{13}\)
• If one of these is the sum of the other two, the points are collinear.

This occurs because if all points lie on a straight line, the largest distance is simply the sum of the two smaller segments. In practice, ensure first you correctly identify the largest distance among the three pairs to make the correct comparison.

Triangle Area Method

The triangle area method is an elegant way to check collinearity by determining the area of the triangle formed by the three given points. If this area equals zero, it implies that the points are on the same line.To find the area of the triangle given points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):

• Use the formula: \[ A = \frac{1}{2} | x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) | \]

Upon calculation, if the area \(A\) is zero, it means the points are collinear, as they cannot form a triangle, just a line.

This method is particularly useful as it avoids potential undefined behaviors or inaccuracies from calculating slopes, and directly gives us a reliable answer to the collinearity of the points.