Math Calculators, Lessons and Formulas
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Given two lines
If is perpendicular to ,then: .
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.
Determine whether the following pairs of lines are parallel.
l1: y = x + 6
l2: the line joining A (1, 4) and B (-4, -1)
Since the two gradients are the same, the pair of lines is parallel.
Given two lines
If is perpendicular to , then: .
y = 3x + 14 is perpendicular to y= x - 72
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.
y - (-1) = (- 3/2)(x - 4)
y + 1 = (- 3/2)x + 6
y = (- 3/2)x + 5
Find the perpendicular bisector of the line segment joining A (-3, 4) and B (2, -1).
Gradient of AB =
Gradient of perpendicular bisector:
Midpoint of AB =