The equations of three lines are given below.
Line 1: $4 x+10 y=2$
Line 2: $2 y=5 x+3$
Line 3: $y=\frac{ 5 }{ 2 } x- 4$
For each pair of lines, determine whether they are parallel, perpendicular, or neither.
Line 1 and Line 2: Parallel Perpendicular Neither
Line 1 and Line 3: Parallel Perpendicular Neither
Line 2 and Line 3: Parallel Perpendicular Neither
Answer (1)
Rewrite each line in slope-intercept form to find their slopes.
Compare the slopes of Line 1 and Line 2: they are perpendicular.
Compare the slopes of Line 1 and Line 3: they are perpendicular.
Compare the slopes of Line 2 and Line 3: they are parallel.
Explanation
Problem Analysis We are given three lines and we need to determine whether each pair of lines is parallel, perpendicular, or neither. To do this, we will first rewrite each line in slope-intercept form ( y = m x + b ), where m is the slope. Then we will compare the slopes of each pair of lines. If the slopes are equal, the lines are parallel. If the product of the slopes is -1, the lines are perpendicular. Otherwise, the lines are neither parallel nor perpendicular.
Finding Slopes Let's rewrite each line in slope-intercept form:
Line 1: 4 x + 10 y = 2 . Subtracting 4 x from both sides gives 10 y = − 4 x + 2 . Dividing both sides by 10 gives y = − 10 4 x + 10 2 , which simplifies to y = − 5 2 x + 5 1 . The slope of Line 1 is m 1 = − 5 2 .
Line 2: 2 y = 5 x + 3 . Dividing both sides by 2 gives y = 2 5 x + 2 3 . The slope of Line 2 is m 2 = 2 5 .
Line 3: y = 2 5 x − 4 . The slope of Line 3 is m 3 = 2 5 .
Comparing Slopes Now let's compare the slopes of each pair of lines:
Line 1 and Line 2: m 1 = − 5 2 and m 2 = 2 5 . Since m 1 = m 2 and m 1 ⋅ m 2 = − 5 2 ⋅ 2 5 = − 1 , Line 1 and Line 2 are perpendicular.
Line 1 and Line 3: m 1 = − 5 2 and m 3 = 2 5 . Since m 1 = m 3 and m 1 ⋅ m 3 = − 5 2 ⋅ 2 5 = − 1 , Line 1 and Line 3 are perpendicular.
Line 2 and Line 3: m 2 = 2 5 and m 3 = 2 5 . Since m 2 = m 3 , Line 2 and Line 3 are parallel.
Conclusion Therefore, Line 1 and Line 2 are perpendicular, Line 1 and Line 3 are perpendicular, and Line 2 and Line 3 are parallel.
Examples
Understanding the relationships between lines (parallel, perpendicular, or neither) is crucial in various real-world applications. For instance, architects use these concepts to design buildings with walls that are either parallel or perpendicular to each other. Similarly, in navigation, understanding the angles and relationships between different paths or routes is essential for determining the direction and avoiding collisions. These concepts also form the basis for coordinate systems used in mapping and GPS technology.
