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Parallel And Perpendicular Lines

Referencing Styles : Harvard | Pages : 1

Parallel lines are the lines in mathematics that are having same slope and it is assumed that the parallel lines will never touch each other. Parallel lines will continuously move in the same direction without touching each other. This is because one of the important assumptions that are mainly helping in drawing parallel lines is that they are originating from the same plane. Parallel lines will go parallel with each other and will never intersect with each other. Another component of parallel line is that parallel lines are having great depth.

The parallel lines are having same slope. For example, Y=2X+3 and Y=2X+1 are two lines that are going parallel to each other as the slope of both the lines is 2. Now parallel lines are having some special characteristics regarding the angles. When the two lines are parallel then the pair of corresponding angles will be equal, or alternative pair of interior angles will be equal, or alternative pair of exterior angles will be equal and last charectistics of parallel line is sum of consecutive interior angles will be equal to 180 degree.



In the above figure, the corresponding angles are a & e which are equal. c & f are the alternative interior angles that is equal. b & g are alternative external angles that are equal. d & f are the sum of consecutive interior angles that are summing up to 180 degree. In order to determine the equation of parallel lines lets us consider the following problems. In order to find the parallel line to the 4x-5y=12 we have to determine the slope of the given line. Y= 4/5x-12/5. In the above equation, the slope of the line is 12 and the m= 4/5. Thus from the given solution, the equation of parallel line is Y=b+mx and all the lines that will be parallel to this one will be having slope of m =4/5. Parallel lines can be of two types- horizontal and vertical parallel lines. In spite of having so many common properties among two lines, the parallel lines will never meet to each other or will never intersect each other. The behaviour of the parallel lines are more or less like railway lines.

Two lines can be said as perpendicular to each other if one line intersects another line in an angle of 90 degree. Now a line is said to be in perpendicular to each other, if the dihedral angle at which they are going to meet is 90 degree.



Now perpendicular lines are to be drawn in a special techniques. The first step that is needed to follow is consider straight line AB. Now pointing the point of compass on P a circle is drawn on AB to get points like A’B’. This circle A’B’ is shown as red. Now two more circles are being drawn from centre A’ and B’ having equal radius. Let P&Q be the point of intersection of these two circles as shown in the diagram above. Now joining P and Q using straight line will definitely give perpendicular where the point O is considered as the foot of perpendicular.

In a two dimensional planes, the right angles can be formed by two intersected lines, if the product of their slopes is equal to -1. Consider two linear equations like y=a1x+b1 and y= a2x+b2. In order to consider these two lines as perpendicular to each other, a1a2 = -1. On the other hand, this method is not at all applicable if the product of slope is zero or undefined. Now in order to identify the equations of perpendicular lines let us consider following example. Let the parent line is 2x-3y=6.
Converting the given equation into slope intercept form is 3y=2x-6 and y= 2/3x-2 and from this the slope of the line can be found that m=2/3. The slope of the perpendicular line will be negative reciprocal in nature. This means the slope of the perpendicular line is -3/2. Putting the value of slope in the equation of y=mx+b will give the perpendicular lines that will meet the straight line at 90 degree. Now this will definitely aiming to draw the perpendicular line. Now it should be kept in mind that perpendicular line, slope must be negative reciprocal in nature.

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