According to Euclid, a straight line goes on forever. When there is more than one line in a plane, the situation becomes more interesting. If two lines never intersect, the lines are parallel. If two lines intersect at a right angle -- 90 degrees -- the lines are said to be perpendicular. The key to understanding how lines relate to each other is the concept of slope, which is the relationship that all lines have to the background plane.
A horizontal line has a slope of zero. If the line is vertical, the slope is said to be undefined. For all other lines, the slope is found by drawing (or imagining) a small right triangle formed by short vertical and horizontal lines where a segment of the line being tested is the hypotenuse. The length of the vertical line divided by the length of the horizontal line is the slope of the line in question.
Parallel lines have the same slope. You do not have to graph the lines and construct the defining triangle to find the slope. If the equation of the line is in the proper form, you can read the slope directly from the formula. The slope form is y = mx + b. Manipulate your formula until it is in this form and "m" is the slope. For example, if your line has the equation Ax - By = C, a little algebraic manipulation puts it in the equivalent form y = (A/B)x - C/B, so the slope of this line is A/B.
The slopes of perpendicular lines have a specific relationship. If the slope of line No. 1 is m, the slope of a line perpendicular to it will have slope -1/m. The slopes of perpendicular lines are negative reciprocals of each other. If the slope of a particular line is 3, all the lines that are perpendicular to the line will have slope -1/3.
Building a Specific Line
Knowing about slopes, parallel lines and perpendicular lines allows you to construct any kind of line through any point. Consider, for example, the problem of finding the equation for a line that goes through the point (3, 4) and is perpendicular to the line 3x + 4y = 5. Manipulating the equation of the known line, you get y = -(3/4)x + 5/4. The slope of this line is -3/4, and the slope of the line perpendicular to this line is 4/3. The perpendicular lines will look like this: y = 4/3x + b. For the line that goes through (3, 4), you can plug in the numbers like this: 4 = 4/3(3) + b, which means that b = 0. The equation for the line that goes through (3, 4) and is perpendicular to the line 3x + 4y = 5 is y = 4/3x or 4x - 3y = 0.
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