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:

  1. The slope of a line passing through two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is given by: m=y2y1x2x1m = \frac{y_2 - y_1}{x_2 - x_1}

  2. Lines are parallel if their slopes are equal.

  3. Lines are perpendicular if the product of their slopes is 1-1 (their slopes are negative reciprocals).


1. Calculate slopes for each line:

Line aa:

Points: (0,4)(0, 4) and (4,3)(4, 3)
ma=3440=14=14m_a = \frac{3 - 4}{4 - 0} = \frac{-1}{4} = -\frac{1}{4}

Line bb:

Points: (0,1)(0, 1) and (4,0)(4, 0)
mb=0140=14=14m_b = \frac{0 - 1}{4 - 0} = \frac{-1}{4} = -\frac{1}{4}

Line cc:

Points: (2,0)(2, 0) and (4,4)(4, 4)
mc=4042=42=2m_c = \frac{4 - 0}{4 - 2} = \frac{4}{2} = 2


2. Check for parallel lines:

Lines are parallel if their slopes are the same.

  • Line aa and Line bb both have slopes m=14m = -\frac{1}{4}.
    So, aa and bb are parallel.

3. Check for perpendicular lines:

Lines are perpendicular if the product of their slopes is 1-1.

  • For aa and cc:
    mamc=(14)2=24=121m_a \cdot m_c = \left(-\frac{1}{4}\right) \cdot 2 = -\frac{2}{4} = -\frac{1}{2} \neq -1 aa and cc are not perpendicular.

  • For bb and cc:
    mbmc=(14)2=24=121m_b \cdot m_c = \left(-\frac{1}{4}\right) \cdot 2 = -\frac{2}{4} = -\frac{1}{2} \neq -1 bb and cc are not perpendicular.

  • For aa and bb:
    mam_a and mbm_b are the same (14-\frac{1}{4}), so they cannot be perpendicular.


Final Answer:

  1. Lines aa and bb are parallel, because they have the same slope 14-\frac{1}{4}.
  2. None of the lines are perpendicular, because no pair of slopes has a product of 1-1.

Do you want further details or have any questions?
Here are 5 related questions to expand this topic:

  1. How do you determine the slope of a vertical line?
  2. Can two lines be neither parallel nor perpendicular?
  3. What does the slope of a line tell you about its steepness?
  4. How do you find the equation of a line given its slope and a point on it?
  5. 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