SLOPE (m) of a line

Remarks:

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.

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