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Chapter 2: Problem 12

Points that lie on the same straight line are said to be collinear. Determine if the following points are collincar. a. \((2,3),(4,7),\) and (8,15) b. \((-3,1),(2,4),\) and (7,8)

Short Answer

Expert verified
a. Yes, collinear. b. No, not collinear.

Step by step solution

01

Understand Collinear Points

Points are collinear if they lie on the same straight line. This can be determined by checking if the slope between each pair of points is the same.
02

Slope Formula

Recall the slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). We will use this to find the slopes between the given points.
03

Calculate Slopes for Points in (a)

For points \( (2,3),(4,7),(8,15) \):Slope between (2,3) and (4,7): \( m_1 = \frac{7 - 3}{4 - 2} = \frac{4}{2} = 2 \)Slope between (4,7) and (8,15): \( m_2 = \frac{15 - 7}{8 - 4} = \frac{8}{4} = 2 \)Since \( m_1 = m_2 \), the points are collinear.
04

Calculate Slopes for Points in (b)

For points \( (-3,1),(2,4),(7,8) \):Slope between (-3,1) and (2,4): \( m_3 = \frac{4 - 1}{2 - (-3)} = \frac{3}{5} \)Slope between (2,4) and (7,8): \( m_4 = \frac{8 - 4}{7 - 2} = \frac{4}{5} \)Since \( m_3 \) is not equal to \( m_4 \), the points are not collinear.

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Key Concepts

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

slope formula
Understanding the slope formula is crucial for determining if points are collinear. The slope describes how steep a line is and is calculated using the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

  • The numerator \( y_2 - y_1 \) represents the difference in the \( y \)-coordinates of two points.
  • The denominator \( x_2 - x_1 \) represents the difference in the \( x \)-coordinates of the same two points.

By comparing the slopes between different pairs of points, we can determine if they lie on the same line. If the slopes are equal, the points are collinear. If not, they are not collinear.
linear equations
A linear equation represents a straight line on a coordinate plane. It can be written in various forms, but one common form is the slope-intercept form: \( y = mx + b \), where \( m \) is the slope and \( b \) is the intercept.

  • The slope \( m \) tells us how steep the line is.
  • The intercept \( b \) is the point where the line crosses the y-axis.

In coordinate geometry, determining if points fit a linear equation helps in establishing collinearity. By checking if the points follow the same linear equation, we can confirm if they lie on a common straight line.
coordinate geometry
Coordinate geometry, also known as analytic geometry, involves using coordinate points to solve geometric problems. It blends algebra and geometry, allowing us to use equations and formulae to derive geometric properties.

  • By plotting points on a coordinate plane, we can visually verify relationships between them.
  • The slope formula and linear equations are tools within coordinate geometry that help in understanding the arrangement of points.
For example, to determine if points are collinear, we utilize the slope formula to calculate the slopes between pairs of points. If the slopes are equal for all pairs, the points lie on the same straight line in the coordinate plane.

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