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Straight Line

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}, d = (y_{2} - y_{1})^{2} + (x_{2} - x_{1})^{2}, m_{1} = - \frac{1}{m _{2}}

p. 37 prints slope-intercept y = mx + b, point-slope y − y1 = m(x − x1), slope m = (y2 − y1)/(x2 − x1), the angle α = arctan[(m2 − m1)/(1 + m1 m2)], perpendicular lines iff m1 = −1/m2, and the distance between two points. The points below are original teaching numbers. Slope 2 is the true slope, not a stand-in for 90°.

m

Slope

d

Distance between two points

α

Angle between two lines

Step-by-step solved example

A line passes through (1, 2) and (4, 8). Find the slope, the distance, and a perpendicular slope.

Equal-scale line through (1, 2) and (4, 8). True slope is 2, not a 90° stand-in. Perpendicular slope is −1/2.

1. Slope
m = (8 − 2)/(4 − 1) = 6/3 = 2. Point-slope: y − 2 = 2(x − 1), so y = 2x.
$m = 2$
2. Distance
d = √[(8−2)² + (4−1)²] = √(36+9) = √45 = 6.708… → 6.71.
$d = 6.71$
3. Perpendicular
A perpendicular slope is −1/m = −1/2. The lines meet at 90°, which is not the same as slope 2.
$m_{⊥} = - 1/2$

Answer: m = 2, d = 6.71, perpendicular slope = −1/2

8 practice questions

0/8 correct

1.Slope through (1, 2) and (4, 8)?

2.Distance between those points, to two decimals?

3.Slope perpendicular to m = 2?

4.Point-slope through (1, 2) with m = 2 is

5.Two lines are perpendicular when

6.The angle between two slopes uses

7.y-intercept of the line through (1, 2) and (4, 8)?

8.If 1 + m1 m2 = 0, the angle formula