Problem 56 - Algebra and Trigonometry

Question

Use slopes to determine whether the given points are collinear (lie on a line). (a) \((1,1),(3,9),(6,21)\) (b) \((-1,3),(1,7),(4,15)\)

Step-by-Step Solution

Answer

(a) Points are collinear; (b) Points are not collinear.

Step 1: Calculate the Slope Between the First Two Points (a)

For points \((1,1)\) and \((3,9)\), use the slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute in the values: \(m = \frac{9 - 1}{3 - 1} = \frac{8}{2} = 4\).

Step 2: Calculate the Slope Between the Last Two Points (a)

For points \((3,9)\) and \((6,21)\), use the slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute in the values: \(m = \frac{21 - 9}{6 - 3} = \frac{12}{3} = 4\).

Step 3: Compare Slopes of Points (a)

The slopes of both segments are equal (\(4\)), indicating all three points \((1,1), (3,9), (6,21)\) lie on the same line.

Step 4: Calculate the Slope Between the First Two Points (b)

For points \((-1,3)\) and \((1,7)\), use the slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute in the values: \(m = \frac{7 - 3}{1 + 1} = \frac{4}{2} = 2\).

Step 5: Calculate the Slope Between the Last Two Points (b)

For points \((1,7)\) and \((4,15)\), use the slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute in the values: \(m = \frac{15 - 7}{4 - 1} = \frac{8}{3}\).

Step 6: Compare Slopes of Points (b)

The slopes of the segments differ (\(2\) and \(\frac{8}{3}\)), indicating the points \((-1,3), (1,7), (4,15)\) do not lie on the same line.

Key Concepts

Slope Formula

The slope formula is a fundamental part of coordinate geometry, used to determine the steepness or incline of a line connecting two points in a plane. It's expressed as:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]where

• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points,
• \(m\) represents the slope of the line.

The formula calculates the ratio of the vertical change (rise) to the horizontal change (run) between the two points. A positive slope means the line ascends as it moves right, while a negative slope means it descends. The slope is zero for horizontal lines and undefined for vertical lines.
The consistency of the slope value across multiple points on a line is a key indicator of collinearity, i.e., whether all points lie on the same line. In practice, compute the slopes between consecutive points and compare their values to verify collinearity.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, merges algebra and geometry to elucidate the geometric attributes of figures via algebraic equations. It depends primarily on the coordinate plane, which is a two-dimensional space defined by a horizontal \(x\)-axis and a vertical \(y\)-axis that intersect at the origin \((0,0)\).

• Points are plotted in this plane using pairs called coordinates \((x, y)\).
• Lines and shapes can be analyzed by converting their geometric configurations into algebraic equations.

In our exercise, we investigate whether points are collinear using slopes. In essence, coordinate geometry helps in transforming these points' locations into understandable constructs such as lines and slopes. This transformation simplifies a complex geometric scenario into one where mathematical tools can be employed effectively.

Linear Equation

A linear equation represents a straight line in a coordinate plane. Its general form is \(y = mx + c\), where:

• \(m\) is the slope of the line, determining its steepness or inclination.
• \(c\) is the y-intercept, which is the point where the line crosses the y-axis.

Linear equations translate into a graphical representation of straight lines. Each solution pair \((x, y)\) in a linear equation lies on the line described by the equation.
When checking for collinearity, calculating slopes helps confirm whether a single linear equation can describe several points. If all points have the same slope between any two pairs, they can be described adequately by one linear equation, confirming they are collinear. Linear equations thus serve as a mathematical backbone to many coordinate geometry applications, providing a straightforward method for understanding complex relationships in a plane.