Problem 54
Question
Determine whether the lines \(l_{1}\) and \(l_{2}\) are parallel, perpendicular, or neither. \(l_{1}\) goes through \((3,6)\) and \((4,9), l_{2}\) goes through \((5,3)\) and \((6,0)\)
Step-by-Step Solution
Verified Answer
The lines are neither parallel nor perpendicular.
1Step 1 - Find the Slope of Line 1 (\(l_1\))
To find the slope of line \(l_1\), use the two points \((3, 6)\) and \((4, 9)\). Use the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\): \[m_1 = \frac{9 - 6}{4 - 3} = \frac{3}{1} = 3\]
2Step 2 - Find the Slope of Line 2 (\(l_2\))
To find the slope of line \(l_2\), use the two points \((5, 3)\) and \((6, 0)\). Using the slope formula again: \[m_2 = \frac{0 - 3}{6 - 5} = \frac{-3}{1} = -3\]
3Step 3 - Compare the Slopes
Compare the slopes of \(l_1\) and \(l_2\): \(m_1 = 3\) and \(m_2 = -3\).
4Step 4 - Determine the Relationship Between the Lines
If the product of the slopes is \(-1\), the lines are perpendicular. If the slopes are equal, the lines are parallel. Otherwise, they are neither. Calculate the product of the slopes: \[3 \times -3 = -9\] Since the product is not \(-1\), and the slopes are not equal, the lines are neither parallel nor perpendicular.
Key Concepts
Slope CalculationParallel LinesPerpendicular LinesLine Comparison
Slope Calculation
Understanding how to calculate the slope of a line is fundamental in determining the relationship between lines. The slope is a measure of how steep a line is.
The formula to find the slope, denoted as \(m\), between two points \((x_1, y_1)\) and \((x_2, y_2)\) on a line is: \[m = \frac{y_2 - y_1}{x_2 - x_1}\] This formula calculates the rate at which \(y\) changes with respect to \(x\) as you move along the line.
For example, for a line passing through points \(3, 6\) and \(4, 9\), the slope calculation would be: \[m = \frac{9 - 6}{4 - 3} = \frac{3}{1} = 3\]
By finding the slope of any two lines, you can easily compare their steepness. This forms the basis for understanding relationships like parallelism and perpendicularity between lines.
The formula to find the slope, denoted as \(m\), between two points \((x_1, y_1)\) and \((x_2, y_2)\) on a line is: \[m = \frac{y_2 - y_1}{x_2 - x_1}\] This formula calculates the rate at which \(y\) changes with respect to \(x\) as you move along the line.
For example, for a line passing through points \(3, 6\) and \(4, 9\), the slope calculation would be: \[m = \frac{9 - 6}{4 - 3} = \frac{3}{1} = 3\]
By finding the slope of any two lines, you can easily compare their steepness. This forms the basis for understanding relationships like parallelism and perpendicularity between lines.
Parallel Lines
When lines are parallel, they will never intersect and they have the same slope.
Let's take two lines characterized by their equations or points: If the slope of the first line \(m_1\) is identical to the slope of the second line \(m_2\), then the lines are parallel: \[m_1 = m_2\]
Using our example, if Line 1 has a slope \(m_1 = 3\) and Line 2 also had a slope of \(3\), then these lines would be parallel.
Parallel lines have many useful properties in geometry and real-world applications, such as in building construction and urban planning where roads might run parallel to maximize space.
Let's take two lines characterized by their equations or points: If the slope of the first line \(m_1\) is identical to the slope of the second line \(m_2\), then the lines are parallel: \[m_1 = m_2\]
Using our example, if Line 1 has a slope \(m_1 = 3\) and Line 2 also had a slope of \(3\), then these lines would be parallel.
Parallel lines have many useful properties in geometry and real-world applications, such as in building construction and urban planning where roads might run parallel to maximize space.
Perpendicular Lines
Perpendicular lines intersect at a right angle (90 degrees) and their slopes have a unique relationship.
The slopes of two perpendicular lines multiply together to be \-1\. Mathematically, if the slope of the first line is \(m_1\) and the slope of the second line is \(m_2\), then: \[m_1 \times m_2 = -1\]
Using our example, if Line 1 has a slope \(m_1 = 3\) and Line 2 has a slope \(m_2 = -3\), then: \[3 \times -3 = -9\]
Since -9 isn't equal to -1, these lines are not perpendicular. Confirming lines are perpendicular is especially useful in geometry to solve problems that involve right angles.
The slopes of two perpendicular lines multiply together to be \-1\. Mathematically, if the slope of the first line is \(m_1\) and the slope of the second line is \(m_2\), then: \[m_1 \times m_2 = -1\]
Using our example, if Line 1 has a slope \(m_1 = 3\) and Line 2 has a slope \(m_2 = -3\), then: \[3 \times -3 = -9\]
Since -9 isn't equal to -1, these lines are not perpendicular. Confirming lines are perpendicular is especially useful in geometry to solve problems that involve right angles.
Line Comparison
Comparing lines involves looking at their slopes to determine if they are parallel, perpendicular, or neither.
First, find the slopes of the given lines. For example, for Line 1 through points \(3,6\) and \(4,9\), the slope is \(3\), and for Line 2 through points \(5,3\) and \(6,0\), the slope is \(-3\).
Next, use the slopes to determine the relationship:
First, find the slopes of the given lines. For example, for Line 1 through points \(3,6\) and \(4,9\), the slope is \(3\), and for Line 2 through points \(5,3\) and \(6,0\), the slope is \(-3\).
Next, use the slopes to determine the relationship:
- If the slopes are equal, the lines are parallel.
- If the product of the slopes is \-1\, the lines are perpendicular.
- If neither condition is met, the lines are neither parallel nor perpendicular.
Other exercises in this chapter
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