Slope and Equations of Lines

Finding and using slope in geometry

Slope and Equations of Lines

Slope Formula

The slope between points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2):

m=y2−y1x2−x1=riserunm = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}}

Types of Slope

Positive slope: Line rises (goes up from left to right)

Negative slope: Line falls (goes down from left to right)

Zero slope: Horizontal line (y=ky = k)

Undefined slope: Vertical line (x=kx = k)

Parallel Lines

Parallel lines have equal slopes: m1=m2m_1 = m_2

Perpendicular Lines

Perpendicular lines have negative reciprocal slopes: m1⋅m2=−1m_1 \cdot m_2 = -1

Or: m2=−1m1m_2 = -\frac{1}{m_1}

Equation Forms

Slope-intercept form: y=mx+by = mx + b

Point-slope form: y−y1=m(x−x1)y - y_1 = m(x - x_1)

Standard form: Ax+By=CAx + By = C

Applications

  • Proving lines are parallel or perpendicular
  • Finding equations of medians, altitudes, perpendicular bisectors
  • Coordinate proofs

📚 Practice Problems

1Problem 1easy

❓ Question:

Find the slope of the line through (2,3)(2, 3) and (6,11)(6, 11).

💡 Show Solution

Use the slope formula: m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

m=11−36−2m = \frac{11 - 3}{6 - 2}

m=84=2m = \frac{8}{4} = 2

Answer: Slope = 22

2Problem 2medium

❓ Question:

Are the lines through (1,2)(1, 2), (3,6)(3, 6) and (0,5)(0, 5), (2,9)(2, 9) parallel, perpendicular, or neither?

💡 Show Solution

Find slope of first line: m1=6−23−1=42=2m_1 = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2

Find slope of second line: m2=9−52−0=42=2m_2 = \frac{9 - 5}{2 - 0} = \frac{4}{2} = 2

Since m1=m2=2m_1 = m_2 = 2, the lines are parallel.

Answer: Parallel

3Problem 3hard

❓ Question:

Find the equation of the line perpendicular to y=3x−2y = 3x - 2 that passes through (6,1)(6, 1).

💡 Show Solution

Step 1: Find the perpendicular slope

Original slope: m1=3m_1 = 3

Perpendicular slope: m2=−13m_2 = -\frac{1}{3}

Step 2: Use point-slope form with (6,1)(6, 1)

y−y1=m(x−x1)y - y_1 = m(x - x_1)

y−1=−13(x−6)y - 1 = -\frac{1}{3}(x - 6)

Step 3: Simplify to slope-intercept form

y−1=−13x+2y - 1 = -\frac{1}{3}x + 2

y=−13x+3y = -\frac{1}{3}x + 3

Answer: y=−13x+3y = -\frac{1}{3}x + 3

🎴

Practice with Flashcards

Review key concepts with our flashcard system

📖

Browse All Topics

Explore other calculus topics