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Chapter 2: Problem 40

Given below are descriptions of two lines. Find the slope of Line 1 and Line 2 . Are each pair of lines parallel, perpendicular or neither? Line 1: Passes through (2,5) and (5,-1) Line 2 : Passes through (-3,7) and (3,-5)

Short Answer

Expert verified
The lines are parallel as both have a slope of -2.

Step by step solution

01

Calculate Slope of Line 1

To find the slope of Line 1, we use the slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). We substitute the given points (2, 5) and (5, -1) into the formula:\[ m_1 = \frac{-1 - 5}{5 - 2} = \frac{-6}{3} = -2 \]Thus, the slope of Line 1 is \(-2\).
02

Calculate Slope of Line 2

For Line 2, use the slope formula with the points (-3, 7) and (3, -5):\[ m_2 = \frac{-5 - 7}{3 + 3} = \frac{-12}{6} = -2 \]Thus, the slope of Line 2 is \(-2\).
03

Determine if Lines are Parallel, Perpendicular, or Neither

Two lines are parallel if they have the same slope. Two lines are perpendicular if the product of their slopes is -1. Since the slopes \(-2\) and \(-2\) from Steps 1 and 2 are equal, the lines are parallel.

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Key Concepts

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

Parallel Lines
Parallel lines are lines in a plane that never intersect or meet, no matter how far they are extended. In the realm of geometry, especially coordinate geometry, parallel lines have a special characteristic related to their slopes.
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In coordinate geometry, lines are represented by their slope, a measure of how steep a line is. For any two lines to be parallel, they must have identical slopes. This means that if the slope of Line 1 is the same as the slope of Line 2, the lines are parallel.
  • Matching Slopes: If lines have the same slope, they do not tilt towards each other, thus they remain parallel.
  • Never Intersect: Parallel lines, by definition, will extend infinitely without crossing.
In the example given, both Line 1 and Line 2 have a slope of \(-2\). Therefore, they are parallel lines as their slopes are equal.
Perpendicular Lines
Perpendicular lines meet at a right angle, forming an "L" shape. In coordinate geometry, two lines are perpendicular if the product of their slopes is \(-1\). This relationship is crucial when determining if two lines intersect at 90 degrees.
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When we calculate the slope of two lines:
  • Special Product: The product of slopes being \(-1\) signifies that two lines are perpendicular.
  • Slopes' Inverse Negatives: If one line's slope is the negative reciprocal of the other's, the lines are perpendicular.
As an example, if Line A has a slope of \(m\) and Line B has a slope of \(-\frac{1}{m}\), Line A and B are perpendicular to each other. In the given solution, the lines have slopes of \(-2\) and \(-2\), and since their product is \(4\) (not \(-1\)), these lines are not perpendicular.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is a branch of geometry where points are defined by their position in the Cartesian plane using coordinates. This field allows the systematic study of geometric figures using numerical coordinates.
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Some fundamental components of coordinate geometry include:
  • Coordinate Plane: Comprised of two perpendicular axes: the X-axis (horizontal) and the Y-axis (vertical).
  • Points and Lines: Points are represented as \((x, y)\) on the plane, and lines showcase relationships through slope and intercepts.
  • Slope Formula: Calculates the steepness of a line - \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
Through coordinate geometry, we can graphically represent and find relationships, such as determining if two lines are parallel or perpendicular by simply using their slopes. This approach provides a powerful method to solve geometric problems using algebra.

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Most popular questions from this chapter

A cell phone company offers two data options for its prepaid phones Pay per use: \(\$ 0.002\) per Kilobyte \((\mathrm{KB})\) used Data Package: \(\$ 5\) for 5 Megabytes ( 5120 Kilobytes) \(+\$ 0.002\) per addition \(\mathrm{KB}\) Assuming you will use less than 5 Megabytes, for what range of use will the data package save you money?

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