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Chapter 2: Problem 53

Determine whether the points lie on a straight line. $$ A(-2,1), B(1,7), \text { and } C(4,13) $$

Short Answer

Expert verified
Yes, the points A\((-2,1)\), B\((1,7)\), and C\((4,13)\) lie on a straight line, as the slopes between A and B, and B and C are both equal to 2.

Step by step solution

01

Find the slope between points A and B

To find the slope between points A and B, use the formula: \( m_{AB} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}\) Use the coordinates of A and B: \( A(-2,1) \) and \( B(1,7) \) \( m_{AB} = \frac{7 - 1}{1 - (-2)} = \frac{6}{3} = 2 \) The slope between points A and B is 2.
02

Find the slope between points B and C

To find the slope between points B and C, use the formula: \( m_{BC} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} \) Use the coordinates of B and C: \( B(1,7) \) and \( C(4,13) \) \( m_{BC} = \frac{13 - 7}{4 - 1} = \frac{6}{3} = 2 \) The slope between points B and C is 2.
03

Compare the slopes

Now we will compare the slopes to determine if the points lie on a straight line. If \(m_{AB} = m_{BC}\), then A, B, and C lie on a straight line. In our case, \(m_{AB} = 2\) and \(m_{BC} = 2\) Since \(m_{AB} = m_{BC}\) , the points A, B, and C lie on a straight line.

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Key Concepts

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

Slope Calculation
In coordinate geometry, the slope is a fundamental concept that quantifies the steepness of a line. To calculate the slope, we use the formula: ewlineewline ewlineewline(\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]ewlineewline ewlineewline) ewlineewline where ewlineewline (\( (x_1, y_1) \)ewlineewline ewlineewline and ewlineewline (\( (x_2, y_2) \)ewlineewline ewlineewline) are the coordinates of two distinct points on the line. The numerator represents the vertical change (rise), while the denominator represents the horizontal change (run) between the two points. For instance, using points A and B from the exercise above, the slope calculation resulted in a value of 2, implying a consistent rise over run ratio.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is a branch of geometry that describes figures and shapes using a coordinate system. This system enables the algebraic representation of geometric shapes, facilitating the analysis of shapes using algebraic equations. For example, the positions of points A, B, and C in the exercise are described using coordinates. Understanding the relationship between these points helps in determining geometric properties such as collinearity, which is whether the points lie on the same straight line. The slopes calculated between pairs of points aid in establishing this relationship.
Straight Line Equation
A straight line in coordinate geometry can be represented algebraically by the linear equation ewlineewline (\[ y = mx + c \]ewlineewline ewlineewline) where ewlineewline (\( m \)ewlineewline ewlineewline) is the slope of the line and ewlineewline (\( c \)ewlineewline ewlineewline) is the y-intercept, the point where the line crosses the y-axis. For points to lie on the same line, they must satisfy the same linear equation; in other words, they must have the same slope when connected pairwise. As shown in the textbook example, the equal slopes ewlineewline (\( m_{AB} \)ewlineewline ewlineewline and ewlineewline (\( m_{BC} \)ewlineewline) confirmed that points A, B, and C lie on a straight line.

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Most popular questions from this chapter

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