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Chapter 2: Problem 58

Show that the points \((-84,-14),(21,1),\) and (98,12) lie on a line.

Short Answer

Expert verified
The slopes between the points \((-84,-14)\) and \((21,1)\), and between the points \((21,1)\) and \((98,12)\) are both equal to \(\frac{1}{7}\). Therefore, the three points lie on the same straight line.

Step by step solution

01

Calculate the slope between the first two points.

We use the formula for the slope of a line given two points: \[m = \frac{y_2 - y_1}{x_2 - x_1}\] So, for the points \((-84,-14)\) and \((21,1)\), we have: \[m_1 = \frac{1 - (-14)}{21 - (-84)}\] Now, calculate the slope between the second and third points, \((21,1)\) and \((98,12)\).
02

Calculate the slope between the second and third points.

Using the same formula for the slope of a line, for points \((21,1)\) and \((98,12)\), we have: \[m_2 = \frac{12 - 1}{98 - 21}\] Now compare the slopes of the line segments to determine if they are equal.
03

Compare the slopes.

\[ m_1 = \frac{1 - (-14)}{21 - (-84)} = \frac{15}{105} \] and \[ m_2 = \frac{12 - 1}{98 - 21} = \frac{11}{77} \] Simplify the fractions: \[ m_1 = \frac{15}{105} = \frac{1}{7} \] and \[ m_2 = \frac{11}{77} = \frac{1}{7} \] We can see that the slopes are equal, i.e., \(m_{1} = m_{2}\).
04

Conclusion

Since the slopes of the line segments are equal, we can conclude that the three points \((-84,-14)\), \((21,1)\), and \((98,12)\) lie on the same straight line.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope of a Line
The slope of a line is a measure of its steepness and direction. It is essential in determining if points are collinear, meaning they lie on the same line. To calculate the slope between two points, \(x_1, y_1\) and \(x_2, y_2\), we use the formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]. This formula provides a ratio that indicates the change in the y-coordinates relative to the change in the x-coordinates.
  • A positive slope indicates that the line rises as it moves from left to right.

  • A negative slope shows the line falls as it goes from left to right.

  • Zero slope means the line is horizontal.

  • An undefined slope is vertical.
In the exercise, we calculate the slope between each pair of points to verify they are collinear. Both segments had the same slope, confirming they lie on the same line.
Coordinate Geometry
Coordinate geometry, also known as analytical geometry, allows us to use algebraic equations to represent geometric principles. When given points on a plane, like the points in the exercise, we can assess their relationships using coordinates.
  • Points are defined by their x and y coordinates.

  • Lines are defined based on the relationship between multiple points.
Using the properties of coordinate geometry, we apply algebraic methods to determine if multiple points are collinear. In our example, by calculating slopes, we demonstrate that the points form a line when plotted in the coordinate system. This analytical approach simplifies geometric problems by reducing them to algebraic expressions.
Linear Equations
A linear equation represents a straight line in geometry and has a general form of \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Linear equations are fundamental when it comes to describing relationships between variables. In our exercise, although we did not express the relationship in equation form, we verified the linearity by equality of slopes.
  • The slope-intercept form is useful for quickly identifying line properties.

  • By knowing the slope and a point, the equation can be constructed.
To find if three points lie on a line, we tested for equal slopes between each pair of points. If they are the same, as shown, the points satisfy the condition for collinearity, which means they fall on the path of a linear equation.

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