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Analytic Geometry: (lesson 3 of 4)

Parallel and Perpendicular Lines

Given two lines

Parallel Lines

If Parallel Lines definitionis perpendicular to Parallel Lines definition,then: Parallel Lines definition.

Example 1:

y = 3x + 14 is parallel to y = 3x - 72

If two lines are not parallel, there exist a point of intersection. This point can be found by solving the two equations simultaneously.

Example2:

Determine whether the following pairs of lines are parallel.

l1: y = x + 6

l2: the line joining A (1, 4) and B (-4, -1)

Solution:

Gradient of Parallel Lines example

Gradient of Parallel Lines example

Since the two gradients are the same, the pair of lines is parallel.

Perpendicular Lines

Given two lines

Perpendicular Lines definition

If Perpendicular Lines definitionis perpendicular to Perpendicular Lines, then: Perpendicular Lines definition.

Example 3:

y = 3x + 14 is perpendicular to y= Perpendicular Lines examplex - 72

Example 4:

Given the line 2x - 3y = 9 and the point (4, -1), find lines through the point that are

1: parallel to the given line and

2: perpendicular to it.

Solution for parallel line:

Clearly, the first thing we need to do is solve "2x - 3y = 9" for "y=", so that we can find the reference slope:

2x - 3y = 9 -3y = -2x + 9 y = (2/3)x - 3

So the reference slope from the reference line is m = 2/3.

Since a parallel line has an identical slope, then the parallel line through (4, -1) will have slope m = 2/3. Hey, now I have a point and a slope! So I'll use the point-slope form to find the line:

y - (-1) = ( 2/3)(x - 4)

y + 1 = ( 2/3)x - 8/3

y = ( 2/3)x - 8/3 - 3/3

y = ( 2/3)x - 11/3

This is the parallel line that they asked for.

Solution for perpendicular line:

For the perpendicular line, we have to find the perpendicular slope. The reference slope is m = 2/3, and, for the perpendicular slope, we'll flip this slope and change the sign. Then the perpendicular slope is m = - 3/2. Now we'll use the slope-intercept form.

Perpendicular Lines example

y - (-1) = (- 3/2)(x - 4)

y + 1 = (- 3/2)x + 6

y = (- 3/2)x + 5

Example 5:

Find the perpendicular bisector of the line segment joining A (-3, 4) and B (2, -1).

Solution:

Gradient of AB = math example

Gradient of perpendicular bisector:

Perpendicular Lines solution

Midpoint of AB = Perpendicular Lines solution

Equation:

Perpendicular Lines equation