Parallel and Perpendicular Lines Revision Notes for Leaving Cert Mathematics

Parallel and Perpendicular Lines

Understanding parallel and perpendicular lines

Parallel lines are lines that never meet and always maintain the same distance between them. The key characteristic of parallel lines is that they have exactly the same slope.

Perpendicular lines are lines that meet at right angles (90°). The slopes of perpendicular lines have a special relationship - they are negative reciprocals of each other.

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Understanding these fundamental definitions is crucial for solving problems involving line relationships. The slope relationship is the key to identifying and working with parallel and perpendicular lines.

Slope relationships

For parallel lines

If two lines are parallel, their slopes are equal:

Line 1: slope = $m_1$
Line 2: slope = $m_2$
If lines are parallel, then $m_1 = m_2$

For perpendicular lines

If two lines are perpendicular, their slopes are negative reciprocals:

Line 1: slope = $m_1$
Line 2: slope = $m_2$
If lines are perpendicular, then $m_1 \times m_2 = -1$ , or $m_2 = -\frac{1}{m_1}$

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Critical Relationship: For perpendicular lines, the product of their slopes always equals -1. This is the fundamental test for perpendicularity.

Finding the slope from the general form

When a line is given in the general form $ax + by + c = 0$ , you can find its slope by rearranging it into the form $y = mx + c$ .

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Method:

Start with $ax + by + c = 0$
Rearrange to get: $by = -ax - c$
Divide by $b$ : $y = \left(-\frac{a}{b}\right)x + \left(-\frac{c}{b}\right)$
The slope is $-\frac{a}{b}$

Finding equations of parallel lines

To find the equation of a line parallel to a given line and passing through a specific point, you need to follow a systematic approach that utilises the fact that parallel lines have identical slopes.

Steps:

Find the slope of the given line
Use the same slope for your new parallel line
Use the point-slope form: $y - y_1 = m(x - x_1)$
Substitute the known point $(x_1, y_1)$ and slope $m$
Simplify to get the final equation

Finding equations of perpendicular lines

To find the equation of a line perpendicular to a given line and passing through a specific point, the key step is calculating the perpendicular slope correctly.

Steps:

Find the slope of the given line
Calculate the perpendicular slope using $m_2 = -\frac{1}{m_1}$
Use the point-slope form: $y - y_1 = m(x - x_1)$
Substitute the known point $(x_1, y_1)$ and perpendicular slope
Simplify to get the final equation

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Common Mistake Alert: When finding the negative reciprocal, remember that if the original slope is $\frac{a}{b}$ , the perpendicular slope is $-\frac{b}{a}$ (flip the fraction AND change the sign).

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Worked Example: Finding a perpendicular line

Question: Find the equation of the line through the point $(-2, 3)$ which is perpendicular to the line $2x - y + 5 = 0$ .

Solution: Step 1: Find the slope of the given line $2x - y + 5 = 0$

Rearrange: $-y = -2x - 5$
Multiply by $-1$ : $y = 2x + 5$
The slope is $2$

Step 2: Find the perpendicular slope

Perpendicular slope = $-\frac{1}{2}$

Step 3: Use point-slope form with $(-2, 3)$ and slope $-\frac{1}{2}$

$y - 3 = -\frac{1}{2}(x + 2)$
$y - 3 = -\frac{x}{2} - 1$
Multiply through by $2$ : $2y - 6 = -x - 2$
Rearrange: $x + 2y - 4 = 0$

Answer: $x + 2y - 4 = 0$

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Worked Example: Identifying parallel and perpendicular lines

Given these line equations:

a: $y = 2x - 3$
b: $y = \frac{1}{2}x + 5$
c: $y = x + 3$
d: $y = -2x - 4$
e: $y = -\frac{1}{2}x + 4$
f: $y = 2x - 2$

Parallel lines: Lines a and f (both have slope $2$ ) Perpendicular lines: Lines b and e (slopes $\frac{1}{2}$ and $-\frac{1}{2}$ are negative reciprocals)

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Worked Example: Reading slope from a graph

From the graph above, we can determine the slope by identifying two points on the line and using the slope formula. The line passes through the origin and the marked point at approximately $(7, 4)$ , giving a slope of $\frac{4}{7}$ .

Exam tips

When working with parallel and perpendicular lines in exams, there are several key strategies that will help you avoid common mistakes and work more efficiently.

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Essential Exam Strategies:

Always check that perpendicular slopes multiply to give $-1$
For parallel lines, simply use the same slope as the given line
When finding slopes from general form $ax + by + c = 0$ , remember the slope is $-\frac{a}{b}$
Practice converting between different forms of line equations
Double-check your arithmetic when calculating negative reciprocals

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Key Points to Remember:

Parallel lines have equal slopes and never meet
Perpendicular lines meet at right angles and have slopes that are negative reciprocals
Slope relationship: If slopes are $m_1$ and $m_2$ , then perpendicular lines satisfy $m_1 \times m_2 = -1$
Point-slope form $y - y_1 = m(x - x_1)$ is essential for finding line equations
Always verify your answer by checking the slope relationship

Parallel and Perpendicular Lines (Leaving Cert Mathematics): Revision Notes