Math Problem Statement
Determine which lines, if any, are parallel or perpendicular. Explain.
Line a passes through $\left(0,\ 4\right)$ and $\left(4,\ 3\right)$ .
Line b passes through $\left(0,\ 1\right)$ and $\left(4,\ 0\right)$ .
Line c passes through $\left(2,\ 0\right)$ and $\left(4,\ 4\right)$ .
are parallel, because they have .
Question 2
are perpendicular, because they have slopes that are .
Solution
To determine which lines are parallel or perpendicular, we calculate the slopes of the lines. Recall:
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The slope of a line passing through two points and is given by:
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Lines are parallel if their slopes are equal.
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Lines are perpendicular if the product of their slopes is (their slopes are negative reciprocals).
1. Calculate slopes for each line:
Line :
Points: and
Line :
Points: and
Line :
Points: and
2. Check for parallel lines:
Lines are parallel if their slopes are the same.
- Line and Line both have slopes .
So, and are parallel.
3. Check for perpendicular lines:
Lines are perpendicular if the product of their slopes is .
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For and :
and are not perpendicular. -
For and :
and are not perpendicular. -
For and :
and are the same (), so they cannot be perpendicular.
Final Answer:
- Lines and are parallel, because they have the same slope .
- None of the lines are perpendicular, because no pair of slopes has a product of .
Do you want further details or have any questions?
Here are 5 related questions to expand this topic:
- How do you determine the slope of a vertical line?
- Can two lines be neither parallel nor perpendicular?
- What does the slope of a line tell you about its steepness?
- How do you find the equation of a line given its slope and a point on it?
- How can you determine if three lines are concurrent?
Tip: Always remember that parallel lines never intersect, while perpendicular lines intersect at a right angle!
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Math Problem Analysis
Mathematical Concepts
Coordinate Geometry
Slopes of Lines
Parallel Lines
Perpendicular Lines
Formulas
Slope formula: m = (y2 - y1) / (x2 - x1)
Condition for parallel lines: Equal slopes
Condition for perpendicular lines: Product of slopes = -1
Theorems
Parallel Lines Theorem
Perpendicular Lines Theorem
Suitable Grade Level
Grades 8-10
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