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    Topics || Problems

    Equation of a Line

    SLOPE (m) of a line

    \(m = \frac{rise}{run}=\frac{y}{x} = \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\)

    lines lines


    Positive Slopes (m > 0). When x is increasing y is also increasing and vice versa. Typically described the line as sloping upwards to the right or downwards to the left.

    Negative Slopes (m < 0). When x is increasing y is decreasing and when x is decreasing y is increasing. Characterized by a line moving downwards to the right or upwards to the left.

    Zero Slopes (m = 0). These are lines parallel to x - axis.

    Undefined Slopes (m = U). These are lines parallel to y - axis.

    Parallel and Perpendicular lines

    Slopes of Parallel Lines. m1 = m2. Two lines are parallel if they have the same slope and conversely.

    Slopes of Perpendicular Lines. m1 = -1/m2. Two lines are perpendicular if and only if their slopes are negative reciprocals.

    Given two lines 1 and 2. Assume that their slopes are negative reciprocal (\( m_1 = \frac{-1}{m_2}\)). Show that the angle of intersection is a right angle. This includes derivation on angles between lines.

    perpendicular lines perpendicular lines

    Assumption: \(m_1 = \frac{-1}{m_2}\)

    Show that: \(\angle C = 90^o\)

    Using the triangle formed. \(\angle B +\angle C = \angle A\)

    \(\tan B = m_2\) and

    \(\tan A = m_1\)

    \(\tan A = \tan (B+C)= m_1\)

    \( \tan (B+C)= \frac{\tan B + \tan C }{1-\tan B \tan C} = \tan A\)

    \( \tan A = \frac{m_2 + \tan C }{1- m_2 \tan C}\)

    \( m_1= \frac{m_2 + \tan C }{1- m_2 \tan C}\)

    \( \tan C = \frac{m_1-m_2}{1+m_2m_1}\)

    But \(m_1 m_2 = -1\) from the assumption

    \( \tan C = \frac{m_1-m_2}{0} = U\)

    Thus \( \angle C\) must be a right angle.

    Therefore, \( \angle C = 90^o\)


    Let A, B, C, a, b are constants

    1. General Form of a line.\(Ax + By + C = 0\)

    2. Slope intercept form of a line \(y = mx + b\)

    3. Point - Slope form of a line \((y - {y_1}) = m(x - {x_2})\)

    4. Two point form \(\frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} = \frac{{y - {y_1}}}{{x - {x_1}}}\)

    5. Intercept form. \(\frac{x}{a} + \frac{y}{b} = 1\)

    a = x - intercept. The value of x when y is zero.

    b = y - intercept. The value of y when x is zero.